The Solution of Embedding Problems in the Framework of GAPs with Applications on Nonlinear PDEs

Abstract

The paper presents a special class of embedding problems whoes solutions are important for the explicit solution of nonlinear partial differential equations. It is shown that these embedding problems are solvable and explicit solutions are given. Not only are the solutions new but also the mathematical framework of their construction which is defined by a nonstandard function theory built over nonstandard algebraical structures, denoted as “GAPs”. These GAPs must not be neither associative nor division algebras, but the corresponding function theories built over them preserve the most important symmetries from the classical complex function theory in a generalized form: a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.

1. Introduction

Except some small areas, at time the nonlinear world is inaccessible for analytic methods, that is, for methods without using numerical algorithms, we only know some explicit solutions of a few nonlinear differential equations (see e.g., [1–13]), and we know precious few about the “embedding’’ of nonlinear structures without any symmetry in higher dimensional structures with symmetry ([14–16], etc.).

This fact legitimates the creation of a new mathematical theory, presented in [17], which allows an “analytic access’’ on wider regions of nonlinearity. It was shown that this theory succeeds in solving nonlinear partial differential equations (concretely the Einstein equations from General Relativity) if the problem defining parameters (concretely the stress energy tensor) have a certain symmetry which can be seen as a wide generalization of the classical symmetry of holomorphy. This new symmetry—in the following denoted as “∗-symmetry’’ or “GAP-symmetry’’ allows the explicit solution of Einstein equations and also of a broad variety of other partial differential equations.

This leads to the following question as considered here: is it possible to embed an arbitrary 𝑛-dimensional nonlinear partial differential equation (which only symmetry is given by smoothness of all coefficient functions) in an 𝑁-dimensional nonlinear partial differential equation with ∗-symmetry, 𝑁≥𝑛. If this holds, we can solve the given 𝑛-dimensional system by embedding this system into the 𝑁-dimensional system, which is solvable on the base of ∗-symmetry. Since embedding of an equation means embedding of the coefficient functions (which usually are given by vector fields (tensor fields)), our question above boils down to the following question:

is it possible to embed an arbitrary 𝑛-dimensional smooth vector field (tensor field) into a “∗-symmetrical’’ 𝑁-dimensional tensor field, 𝑁≥𝑛?

This question will be answered positively here.

For a better understanding of the practical meaning of this result, some analogies to the classical situation of holomorphy are given (see the following section and also more details in [17]).

(i)The simplest GAP is defined by the two-dimensional field ℂ of the complex numbers. In this case, the ∗-symmetry is defined by the classical symmetry of holomorphy. It is a well-known fact, that an arbitrary smooth real-valued function 𝑓∶𝐼→ℝ, 𝐼⊆ℝ, can be “embedded’’ in a complex valued function 𝐹∶𝐺→ℂ, 𝐺⊆ℂ in the sense that the real part of 𝐹 for values in the region 𝐼 is identically with 𝑓. Our Embedding Theorem is a generalization of this classical result, by generalizing the two-dimensional field ℂ to an 𝑛-dimensional GAP (generally a nonassociative, noncommtative, nondivision algebra).(ii)The classical symmetry of holomorphy leads immediately to related (partly equivalent) symmetries: the well-known Cauchy-Riemannian differential equations and the well-known Cauchy Integral Theorem. The GAP-formalism generalizes these symmetries: it is shown that in GAPs, analogies of the Cauchy-Riemannian differential equations exist (in the following denoted as Pseudo Cauchy-Riemannian differential equations or shorter: “PCRE’’) as well as analogies of the Cauchy Integral Theorem (in the following denoted as Pseudo Cauchy Integral Theorem or shorter: “PCIT’’).(iii)The classical symmetries: Cauchy-Riemannian differential equations and Cauchy’s Integral Theorem lead to a solution theory of the two-dimensional linear Laplace equation. In a similar way, their analogies—the PCRE and the PCIT—lead to a solution theory of 𝑛-dimensional nonlinear partial differential equations.

After these considerations, some remarks to the structure of the paper are presented as follows.

(1)In Section 2.1 the concept of “∗-analyticity’’ is introduced: we will define the so-called ∗-analytic tensor fields by demanding that these fields allow a representation as generalized power series in very general algebras.(2)In Section 2.2 we will specialize these algebras to the so-called “PAk-structures’’ by introducing symmetries which can be seen as wide generalizations/modifications of the classical associativity symmetry. In the function theories built up over special PAk-structures (so-called “pseudorings’’), there exist generalizations of the well-known Cauchy-Riemannian differential equations as well as generalizations of the well-known Cauchy Integral Theorem.(3)In Section 2.3 a further generalization is given by generalizing the PAk-structures to the so-called “GAPs.’’ It is shown that also in this wide framework the main results of Section 2.2 hold.(4)Section 2.4 solves the problem of calculating GAPs explicitly.(5)Section 3 presents applications of the GAP-formalism on famous partial differential equations from Theoretical Physics: new explicit solutions of Einstein equations from General Relativity and Navier-Stokes equations are given. The request of a further generalization of these results (which would be important from a physical point of view) shows the necessity of “embedding low dimensional unsymmetrical structures into higher dimensional symmetrical structures,’’ that is, the necessity of an Embedding formalism, which is developed in the following Section 4.(6)Section 4 presents the Embedding Theorem: it is shown that an arbitrary 𝑛-dimensional smooth vector field always can be embedded into a special 𝑁-dimensional smooth vector field (𝑁≥𝑛) restricted by special GAP-symmetries. In other words, The world of GAPs is wide enough to allow embedding for rather general (smooth) structures. It is remarkable that the Embedding Theorem not only shows the possibility of Embedding but also endows the tools for practical applications. A simple example is given.

Finally some remarks to the style of the presented paper, which is given by accentuation of constructiveness, are presented like it is demanded by the concrete problems: not only are statements to existence and uniqueness desired and given but furthermore the explicit construction of a wide variety of solutions. For this reasons, this paper is not written for pure mathematicians but for physicists and applied mathematicians.

2. Elements of GAP-Theory

In this chapter the most important mathematical concepts will be presented as necessary for our solution method. Some of these concepts have been presented in [17], where the interested reader can find the proofs as missing here. The larger part of concepts is new; the corresponding propositions of course all will be proved in the following.

2.1. The Concept of ∗-Analyticity in General Algebras

We start by remembering on same elementary terms of algebra theory: let 𝑉 denote an arbitrary vector space built over the field ℝ of real numbers, and ∗∶𝑉×𝑉→𝑉 an arbitrary binary distributive operation. The algebraic structure defined by 𝑉 and ∗ will be denoted here as (𝑉;∗). The ∗-operation on 𝑉-vectors can be described by means of an arbitrary 𝑉-base {𝑏_𝑖}, 𝑖=1,…, dim𝑉 according to 𝑏_𝑖∗𝑏_𝑗=𝐶^𝑘_{(∗)𝑖𝑗}𝑏_𝑘, where 𝐶^𝑘_{(∗)𝑖𝑗}∈ℝ denote the structure constants of the algebra (𝑉;∗) with respect to the base {𝑏_𝑖}. Instead of 𝑖=1,…,dim𝑉, we will write in the future 𝑖∈𝐼_1,dim𝑉. The associated base-independent object to the structure constants 𝐶^𝑘_{(∗)𝑖𝑗}, 𝑖,𝑗,𝑘∈𝐼_1,dim𝑉 is the structure constant tensor 𝐶_(∗) of the algebra: 𝐶_(∗)=𝐶^𝑘_{(∗)𝑖𝑗}𝑏^𝑖⊗𝑏^𝑗⊗𝑏_𝑘, with 𝑏^𝑖∈𝑉^∗, where 𝑉^∗ denotes the dual space of 𝑉 and {𝑏^𝑖} the dual base of {𝑏_𝑗}, defined by 𝑏^𝑖(𝑏_𝑗)=𝛿^𝑖_𝑗. By means of 𝐶_(∗) the product 𝑥∗𝑦,𝑥,𝑦∈𝑉 can be developed as follows: 𝑥∗𝑦=(𝑥∗𝑦)^𝑘𝑏_𝑘=(𝑥^𝑖𝑏_𝑖)∗(𝑦^𝑗𝑏_𝑗)=^{distributivity}𝑥^𝑖𝑦^𝑗(𝑏_𝑖∗𝑏_𝑗)=^{definitionof𝐶(∗)}𝑥^𝑖𝑦^𝑗𝐶^𝑘_{(∗)𝑖𝑗}𝑏_𝑘⇒(𝑥∗𝑦)^𝑘=𝑥^𝑖𝐶^𝑘_{(∗)𝑖𝑗}𝑦^𝑗,𝑘∈𝐼_1,dim𝑉. (2.1) In the future we will denote the quadratic matrix 𝑥^𝑖𝐶^𝑘_{(∗)𝑖𝑗} by (𝑥∗)^𝑘_𝑗 and the quadratic matrix 𝐶^𝑘_{(∗)𝑖𝑗}𝑦^𝑗 by (∗𝑦)^𝑘_𝑖 or shorter in formal denotation 𝑥∗ and ∗𝑦. If rank𝑥∗<dim𝑉 for a fixed element 𝑥, then equation 𝑥∗𝑦=0 has a nonzero solution 𝑦∈𝑉 and if rank∗𝑦<dim𝑉 for a fixed element 𝑦, then equation 𝑥∗𝑦=0 has a nonzero solution 𝑥∈𝑉. We remember that in the special case of a ring operation ∗ a nonzero element 𝑥 with 𝑥∗𝑦=0,𝑦 another nonzero element, is called a zero divisor, and a ring without zero divisors is called a division ring. In this work we will use the denotation “zero divisor’’ also for more general algebras in the sense above.

If (𝑉;∗) has a left unit element (shorter: “left unit”), we denote this element as 𝑒_(∗,𝐿), defined by 𝑒_(∗,𝐿)∗𝑥=𝑥,forall𝑥∈𝑉. If (𝑉;∗) has a right unit element (shorter: “right unit”), we denote this element as 𝑒_(∗,𝑅), defined by 𝑥∗𝑒_(∗,𝑅)=𝑥,forall𝑥∈𝑉. If left unit and right unit are identical we will write 𝑒_(∗). The existence of a right unit or a left unit allows to define the following generalizations of the inverse element conception: 𝑥∗𝑥^{−1(∗,𝑅,𝑅)}=𝑒_(∗,𝑅),𝑥^{−1(∗,𝐿,𝑅)}∗𝑥=𝑒_(∗,𝑅),𝑥∗𝑥^{−1(∗,𝑅,𝐿)}=𝑒_(∗,𝐿),𝑥^{−1(∗,𝐿,𝐿)}∗𝑥=𝑒_(∗,𝐿). (2.2) Here the symbol 𝑥^{−1(∗,𝑅,𝑅)} denotes the right inverse of 𝑥 in respect of the right unit 𝑒_(∗,𝑅) (see the superscripts 𝑅,𝑅), 𝑥^{−1(∗,𝐿,𝑅)} denotes the left inverse of 𝑥 in respect of the right unit 𝑒_(∗,𝑅) (see the superscripts 𝐿,𝑅), and so forth. The existence of a unit element 𝑒_(∗)=𝑒_(∗,𝑅)=𝑒_(∗,𝐿) allows the definition of an inverse element 𝑥^−1(∗) for 𝑥∈𝑉: 𝑥^−1(∗)∗𝑥=𝑥∗𝑥^−1(∗)=𝑒_(∗). The extension (∗) in 𝑥^−1(∗) will help us to distinguish an inverse vector from the inverse 𝐴⁻¹ of a quadratic matrix 𝐴.

The classical algebra symmetries (anti)commutativity, associativity, Jacobi-symmetry, existence of a left unit/right unit can be described by means of the structure constants as shown in Table 1.

Table 1 

The 𝐶_(∗) symmetries follow immediately from representing the vectors 𝑥,𝑦,𝑧 in respect of a vector base {𝑏_𝑖} and by calculating the base products 𝑏_𝑖∗𝑏_𝑗,(𝑏_𝑖∗𝑏_𝑗)∗𝑏_𝑘, and so forth, by the structure constants.

Let 𝑉^𝑝∶=𝑉⊗𝑉⋯⊗𝑉 (𝑝 times) denote the 𝑝-repeated tensor product of the vector space 𝑉 and 𝑉_𝑞∶=𝑉^∗⊗𝑉^∗⋯⊗𝑉^∗ (𝑞 times) the 𝑞-repeated tensor product of the dual space 𝑉^∗. With 𝑏_𝑖1⊗𝑏_𝑖2⋯⊗𝑏_𝑖𝑝∈𝑉^𝑝, we denote an arbitrary 𝑉^𝑝-base, and with 𝑏^𝑖1⊗𝑏^𝑖2⋯⊗𝑏^𝑖𝑞∈𝑉_𝑞 the corresponding dual base. Then a tensor 𝑇∈𝑉^𝑝 (also called as “tensor of type (𝑝,0)’’) has a representation 𝑇=𝑇^{𝑖1,…,𝑖𝑝}𝑏_𝑖1⊗𝑏_𝑖2⋯⊗𝑏_𝑖𝑝 with 𝑇^{𝑖1,…,𝑖𝑝}=𝑇(𝑏^𝑖1,𝑏^𝑖2,…,𝑏^𝑖𝑝) and a tensor 𝑆∈𝑉_𝑞 (also called as “tensor of type (0,𝑞)’’) has a representation 𝑆=𝑆_{𝑖1,…,𝑖𝑞}𝑏^𝑖1⊗𝑏^𝑖2⋯⊗𝑏^𝑖𝑞 with 𝑆_{𝑖1,…,𝑖𝑞}=𝑆(𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑞).

After these well-known assumptions, we will start with GAP-Theory by introducing two new product operations 𝑉^𝑝×𝑉→𝑉^𝑝 and 𝑉_𝑞×𝑉→𝑉_𝑞 as follows.

Definition 2.1. Given an arbitrary algebra (𝑉;∗) with structure constant tensor 𝐶_(∗). Then the∗-associatedproduct of order 𝑝: ⊛_(∗)∶𝑉^𝑝×𝑉→𝑉^𝑝 is defined by (𝑏_𝑖1⊗𝑏_𝑖2⋯⊗𝑏_𝑖𝑝)⊛_(∗)𝑏_𝑗∶=𝐶^𝑘_{(∗)𝑖1𝑗}𝑏_𝑘⊗𝑏_𝑖2⋯⊗𝑏_𝑖𝑝+𝑏_𝑖1⊗𝐶^𝑘_{(∗)𝑖2𝑗}𝑏_𝑘⊗𝑏_𝑖3⋯⊗𝑏_𝑖𝑝+⋯+𝑏_𝑖1⊗𝑏_𝑖2⋯⊗𝑏_𝑖𝑝−1⊗𝐶^𝑘_{(∗)𝑖𝑝𝑗}𝑏_𝑘, (2.3) and the∗-dual associated product of order q: ⊛^dual_(∗)∶𝑉_𝑞×𝑉→𝑉_𝑞 by (𝑏^𝑖1⊗𝑏^𝑖2⋯⊗𝑏^𝑖𝑞)⊛^dual_(∗)𝑏_𝑗∶=𝐶^𝑖1_{(∗)𝑘𝑗}𝑏^𝑘⊗𝑏^𝑖2⋯⊗𝑏^𝑖𝑞+𝑏^𝑖1⊗𝐶^𝑖2_{(∗)𝑘𝑗}𝑏^𝑘⊗𝑏^𝑖3⋯⊗𝑏^𝑖𝑞+⋯+𝑏^𝑖1⊗⋯⊗𝑏^𝑖𝑞−1⊗𝐶^𝑖𝑞_{(∗)𝑘𝑗}𝑏^𝑘. (2.4)

It is necessary to write ⊛_(∗) instead of ⊛, because later we will deal with different product operations ∗_𝑘, 𝑘=1,2,… and with the corresponding ∗_𝑘-associated operations ⊛_(∗𝑘). In the case 𝑝=1, it holds by definition that ⊛_(∗)=∗; in the case 𝑞=1, we write also ∗^dual for ⊛^dual_(∗). Due to the linearity of the vector spaces 𝑉^𝑝 and 𝑉_𝑞, the ⊛-products of the base vectors uniquely define products of the form 𝑇⊛_(∗)𝑧 and 𝑆⊛^dual_(∗)𝑧 for arbitrary tensors 𝑇∈𝑉^𝑝, 𝑆∈𝑉_𝑞, and 𝑧∈𝑉. For this we look at the following results:

Proposition 2.2. The components of products 𝑇⊛_(∗)𝑧,𝑆⊛^dual𝑧, 𝑇∈𝑉^𝑝, 𝑆∈𝑉_𝑞, and 𝑧∈𝑉 are given by (𝑇⊛_(∗)𝑧)^{𝑖1𝑖2⋯𝑖𝑝}=(𝑇^{𝑘𝑖2⋯𝑖𝑝}𝐶^𝑖1_{(∗)𝑘𝑗}+𝑇^{𝑖1𝑘𝑖3⋯𝑖𝑝}𝐶^𝑖2_{(∗)𝑘𝑗}+⋯+𝑇^{𝑖1𝑖2⋯𝑖𝑝−1𝑘}𝐶^𝑖𝑝_{(∗)𝑘𝑗})𝑧^𝑗,(𝑆⊛^dual_(∗)𝑧)_{𝑖1𝑖2⋯𝑖𝑞}=(𝑆_{𝑘𝑖2⋯𝑖𝑞}𝐶^𝑘_{(∗)𝑖1𝑗}+𝑆_{𝑖1𝑘𝑖3⋯𝑖𝑞}𝐶^𝑘_{(∗)𝑖2𝑗}+⋯+𝑆_{𝑖1𝑖2⋯𝑖𝑞−1𝑘}𝐶^𝑘_{(∗)𝑖𝑞𝑗})𝑧^𝑗. (2.5) The following algebraical symmetries hold:(𝑇₁+𝑇₂)⊛_(∗)𝑧=(𝑇₁⊛_(∗)𝑧)+(𝑇₂⊛_(∗)𝑧),(𝑇₁⊗𝑇₂)⊛_(∗)𝑧=(𝑇₁⊛_(∗)𝑧)⊗𝑇₂+𝑇₁⊗(𝑇₂⊛_(∗)𝑧),(anti)symmetryof𝑇⇔(anti)symmetryof𝑇⊛_(∗)𝑧,𝑇⊛_(∗)𝑒_(∗,𝑅)=𝑝𝑇, (2.6a)(2.6b)(2.6c)(2.6d)Relation (2.32a) shows a very simple characterization of the ∗-derivative by the Lie-derivative, but it is only possible in a pseudoring with a right unit.

After the partial derivatives, covariant derivatives and Lie-derivatives we will study the exterior derivatives of ∗-analytic forms. Let us denote the set of all in region 𝑊⊆ℝ^𝑛 smooth 𝑞 -forms with 𝔄^𝑞(𝑊), and the subset of all in 𝑊∗-analytical 𝑞-forms with 𝔄^𝑞(𝑊;∗)_(𝑧0).

Proposition 2.15. Given a pseudoring (ℝ^𝑛;∗) of first type, an arbitrary nontrivial 𝑞-form 𝜔∈𝔄^𝑞(𝑊;∗)_(𝑧0),and arbitrary vector fields 𝑢,𝑣∈𝔗¹(𝑊;∗)_(𝑧0),𝑤∈𝔗₁(𝑊;∗)_(𝑧0),𝑣 not constant (“Not constant” means of course not constant in the considered coordinate system), then it holds that 𝑑⋀𝜔=0⇔(ℝ^𝑛;∗)iscommutative,𝑢∗𝑣∈𝔗¹(𝑊;∗)_(𝑧0)⇔(ℝ^𝑛;∗)iscommutative,𝑤∗^dual𝑣∈𝔗₁(𝑊;∗)_(𝑧0)⇔(ℝ^𝑛𝑢;∗)iscommutative,_−1(∗)∈𝔗¹(𝑊;∗)_(𝑧0)⇔(ℝ^𝑛;∗)iscommutative. (2.33)

This proposition shows that some fundamental symmetries with practical importance only exist in commutative pseudorings: the ∗-product of ∗-analytical vector fields is ∗-analytic only for commutative pseudorings, and an analogue result holds for the ∗^dual-product. Also the inverse element of a ∗-analytical vector field will be ∗-analytical only in a commutative pseudoring. Let us sum up the relations for the classical derivatives as follows:

Pseudo-Cauchy-Riemann equations:

𝜕𝑇=𝔇_(∗)𝑇⊛_(∗),𝜕𝑆=𝔇_(∗)𝑆⊛^dual_(∗), (2.34a) Covariant derivatives:

𝐷𝑇=𝔇_(∗)𝑇⊛_(∗)+𝑇⊛_(Γ),𝐷𝑆=𝔇_(∗)𝑆⊛^dual_(∗)−𝑆⊛^dual_(Γ), (2.34b) Lie-derivative:

𝔏_𝑣𝑇=𝔇_(∗)𝑇⊛_(∗)𝑣−𝑇⊛_(∘𝑇)𝔇_(∘)𝑣,𝔏_𝑣𝑆=𝔇_(∗)𝑆⊛^dual_(∗)𝑣+𝑆⊛^dual_(∘𝑇)𝔇_(∘)𝔇𝑣,_(∗)𝑇=𝔏_{𝑒(∗,𝑅)}𝑇, (2.34c) Exterior  derivative: 𝑑⋀𝜔=0⇔(ℝ^𝑛;∗)iscommutative,𝜔∈𝔄^𝑞(𝑊;∗)_(𝑧0), (2.34d)with (ℝ^𝑛;∗),(ℝ^𝑛;∘) pseudorings of first type, 𝑇∈𝔗^𝑝(𝑊;∗)_(𝑧0),𝑆∈𝔗_𝑞(𝑊;∗)_(𝑧0),𝑊⊂ℝ^𝑛,𝑣∈𝔗¹(𝑊;∘)_(𝑧0). We sum up in the following paragraph.

Our aim was to study ∗-analytical tensor fields in the framework of pseudorings, which have been introduced here as a fundamental concept. It was shown that pseudoringsymmetry is an algebraic symmetry which also appears in the world of differential geometry, moreover: the geometrical world prefers the pseudoringsymmetry against the classical symmetry of associativity. For this the study of pseudorings is legitimized, and so the study of tensor fields built over pseudorings. In particular we have studied the partial derivatives of ∗-analytical tensor fields, their covariant derivatives, their Lie-derivatives and their exterior derivatives, all this in the frame of pseudoringsymmetry. It was shown, that all these fundamental derivatives can be written in terms of the so-called ∗-derivative, which has been introduced here as a fundamental concept, following from the symmetry of ∗-analyticity.

2.3. Generalizations. The World of GAPs

Until now we have dealt with PAk-structures (ℝ^𝑛;∗₁,∗₂) and pseudorings (ℝ^𝑛;∗), that is, with algebraic structures defined by maximal two product operations. For some later applications, this framework will be too small and shall be generalized here (The most applications on mathematical physics only need some small parts of this section: the concept of a GAP, of a GAP characteristic, and of GAP Exponentials). Such an algebra generalization will lead us to the concept of GAPs (PAk-structures and pseudorings will be shown as the most simple GAP-structures) and will allow us to generalize the function theoretical concepts of ∗-analyticity, ∗-derivative, ∗-integration, and so forth, from the section above by replacing the underlying PAk-structures (pseudorings) by GAPs.

The aim of this section is to generalize the concepts of PAk-structures, ∗-analyticity, ∗-derivative, ∗-integration, Pseudo Cauchy Riemann equations, and so forth.

Let us denote 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) as an algebraic structure with the operations ∗_𝛼∶ℝ^𝑛×ℝ^𝑛→ℝ^𝑛,𝛼∈𝐼_1,𝑀,𝑀 an arbitrary natural number which also might be infinite. To generalize the concept of ∗-analyticity we have to generalize the concept of the power function from Section 2.1. For this we introduce the maps 𝑃^[𝛼]_{(∗1,∗2,…,∗𝛼)}∶(ℝ^𝑛)^𝑝×(ℝ^𝑛)→(ℝ^𝑛)^𝑝 and 𝑃^dual[𝛼]_{(∗1,∗2,…,∗𝛼)}∶(ℝ^𝑛)_𝑞×(ℝ^𝑛)→(ℝ^𝑛)_𝑞 as follows:

𝑃^[𝛼]_{(∗1,∗2,…,∗𝛼)}(𝐴,𝑧)∶=(⋯((𝐴⊛_(∗𝛼)𝑧)⊛_{(∗𝛼−1)}𝑧)⋯⊛_(∗2)𝑧)⊛_(∗1)𝑃𝑧,𝛼-times,_{dual[𝛼](∗1,∗2,…,∗𝛼)}(𝐵,𝑧)∶=(⋯((𝐵⊛^dual_(∗𝛼)𝑧)⊛^dual_{(∗𝛼−1)}𝑧)⋯⊛^dual_(∗2)𝑧)⊛^dual_(∗1)𝑧,𝛼-times, (2.35)𝐴∈(ℝ^𝑛)^𝑝,𝐵∈(ℝ^𝑛)_𝑞 arbitrary constants, 𝑧∈ℝ^𝑛. After these assumption we can introduce the concept of “chain-analyticity”.

Definition 2.16. Given an arbitrary algebraic structure 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) and arbitrary tensors 𝑇∈𝔗^𝑝(𝑊₁),𝑆∈𝔗_𝑞(𝑊₁), we denote 𝑇 and 𝑆, respectively, as chain-analyticin region 𝑊₁, if there exists an element 𝑧₀∈𝑊₁, which allows the following representations of 𝑇, respectively, 𝑆: 𝑇^{𝑖1⋯𝑖𝑝}(𝑧)={𝐴_[0]+^𝑀_𝛼=1𝑃^[𝛼]_{(∗1,∗2,…,∗𝛼)}(𝐴_[𝛼],𝑧−𝑧₀)}^{𝑖1⋯𝑖𝑝}∶=𝐴_[0]+𝐴_[1]⊛_(∗1)(𝑧−𝑧₀)+(𝐴_[2]⊛_(∗2)(𝑧−𝑧₀))⊛_(∗1)(𝑧−𝑧₀)+((𝐴_[3]⊛_(∗3)(𝑧−𝑧₀))⊛_(∗2)(𝑧−𝑧₀))⊛_(∗1)(𝑧−𝑧₀𝑆)+⋯,_{𝑖1⋯𝑖𝑞}(𝑧)={𝐵_[0]+^𝑀_𝛼=1𝑃^dual[𝛼]_{(∗1,∗2,…,∗𝛼)}(𝐵_[𝛼],𝑧−𝑧₀)}_{𝑖1⋯𝑖𝑞}∶=𝐵_[0]+𝐵_[1]⊛^dual_(∗1)(𝑧−𝑧₀)+(𝐵_[2]⊛^dual_(∗2)(𝑧−𝑧₀))⊛^dual_(∗1)(𝑧−𝑧₀)+((𝐵_[3]⊛^dual_(∗3)(𝑧−𝑧₀))⊛^dual_(∗2)(𝑧−𝑧₀))⊛^dual_(∗1)(𝑧−𝑧₀)+⋯, (2.36)𝑧∈𝑊₁,𝐴_[𝛼] a constant of type (𝑝,0),𝐵_[𝛼] a constant of type (0,𝑞). The set of all in region 𝑊₁ chain-analytical tensors 𝑇 and 𝑆 for fixed 𝑧₀ is denoted as 𝔗^𝑝(𝑊₁;∗₁,∗₂,…,∗_𝑀)_(𝑧0) and 𝔗_𝑞(𝑊₁;∗₁,∗₂,…,∗_𝑀)_(𝑧0).

Consider that in the series above an infinite value of 𝑀 is allowed and an infinite set of ∗_𝛼-operations is allowed to contain an infinite subset of identical operations. In the case ∗₁=∗₂=⋯=∗_𝑀 (=finite or infinite), we will denote the structures above as ∗-analytical as in the earlier case of a single product operation. The denotation “chain-analytical” comes from the fact, that the series above are built by ∗_𝛼-operations like the “elements of a chain”. We see, that for different ∗_𝛼-operations the structure of chain-analytical tensor fields is much more general than the structure of simple ∗-analytical tensor fields. Now we will generalize the concept of PAk-structures for vector spaces with different dimensions.

Now I will present a concept, which allows a short overview about all symmetries—PAk-symmetry as well as standard symmetries—of a given algebraic structure. For this we assign the algebraic structure 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) a 𝑀×𝑀-matrix 𝐺_(𝔛), denoted as the symmetry characteristic of the algebra 𝔛 as follows:

𝐺_{(𝔛)𝛼𝛽}∶=𝑘ifthesubalgebra(ℝ^𝑛;∗_𝛼,∗_𝛽0)isaPAk-structure,otherwise. (2.37) (“0” means: “no PAk symmetry is assumed” and not: “no PAk-symmetry is allowed”. With other words: In the case “0” it is possible that a PAk-symmetry exists, but it is not ensured.)

Definition 2.17. An algebraic structure 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀)is called a GAP of order M, if the symmetry characteristic 𝐺_(𝔛) includes at least one value 1 or 2. In this case 𝐺_(𝔛) is denoted as the GAP characteristic of the GAP 𝔛.

In other words a GAP is an algebraic structure which has at least one PA1-symmetry or one PA2 -symmetry. For further applications it will be advantageable, to generalize the GAP-characteristic by giving an overview not only about the PAk-symmetries but also about the ”standard symmetries” of all algebraic structures (ℝ^𝑛;∗_𝛼), that is, of its possible (anti)commutativity, Jacobi-symmetry, associativity, existence of a right unit 𝑒_(∗,𝑅), existence of a left unit 𝑒_(∗,𝐿), and so forth. Since these symmetries all are defined for a single algebra operation, they only will appear in the main diagonal of 𝐺_(𝔛). Concretely we introduce the following denotations as subscripts of the elements 𝐺_{(𝔛)𝛼𝛼}:

𝑒𝑐∼commutativity,ac∼anticommutativity,𝐽∼Jacobi-symmetry,_(𝑅)𝑒∼existenceofarightunit,_(𝐿)∼existenceofaleftunit,𝑒∼existenceofaunit. (2.38) As an example, we consider an algebraic structure 𝔛=(ℝ^𝑛;∗₁,∗₂,∗₃,∗₄) with the following symmetry characteristic:

𝐺_(𝔛)=010000_𝑒(𝑅)10000_ac,𝐽00200 (2.39) which means that the algebra (ℝ^𝑛;∗₂) has a right unit, the algebra (ℝ^𝑛;∗₃) is anticommutative and satisfies the Jacobi-symmetry (i.e., (ℝ^𝑛;∗₃) is a Lie-algebra), the algebra (ℝ^𝑛;∗₄) is commutative, the algebraic structure (ℝ^𝑛;∗₁,∗₂) is a PA1-structure, and the algebraic structure (ℝ^𝑛;∗₄,∗₂) is a PA2-structure. Per definition 𝔛 is a GAP.

Now some special types of GAPs will be introduced which allow a wide generalization of the functional theoretical concepts of ∗-analyticity, ∗-derivative, Pseudo Cauchy Riemann equations, ∗-integration, and so forth.

Definition 2.18. A PA1-chain is a GAP 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) with PAk_𝛼𝛼−1=1,𝛼∈𝐼_2,𝑀. A PA1-chain is denoted as a closed PA1-chain, if additionally to the above it holds PAk_1𝑀=1. A closed PA1-chain of Lie-typeis defined as a closed PA1-chain of order 3 with the additional restriction (𝑥∗₃𝑦)∗₂𝑧+(𝑦∗₂𝑧)∗₁𝑥+(𝑧∗₁𝑥)∗₃𝑦=0,∀𝑥,𝑦,𝑧∈ℝ^𝑛. (2.40)

The concept of a PA1-chain allows a wide generalization of the earlier concept of the ∗-derivation, which will be presented now.

Definition 2.19. Given a PA1-chain 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) and chain-analytic tensors 𝑇∈𝔗^𝑝(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0),𝑆∈𝔗_𝑞(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0), then the following series expansions (𝔇_(∗1)𝑇)^{𝑖1⋯𝑖𝑝}(𝑧)∶={𝐴_[1]+^𝑀_𝛼≥2𝑃^[𝛼−1]_{(∗2,…,∗𝛼)}(𝛼𝐴_[𝛼],𝑧−𝑧₀)}^{𝑖1⋯𝑖𝑝}∶={𝐴_[1]+2𝐴_[2]⊛_(∗2)(𝑧−𝑧₀)+(3𝐴_[3]⊛_(∗3)(𝑧−𝑧₀))⊛_(∗2)(𝑧−𝑧₀)+⋯}^{𝑖1⋯𝑖𝑝},(𝔇_(∗1)𝑆)_{𝑖1⋯𝑖𝑞}(𝑧)∶={𝐵_[1]+^𝑀_𝛼≥2𝑃^{dual[𝛼−1]}_{(∗2,…,∗𝛼)}(𝛼𝐵_[𝛼],𝑧−𝑧₀)}_{𝑖1⋯𝑖𝑞}∶={𝐵_[1]+2𝐵_[2]⊛^dual_(∗2)(𝑧−𝑧₀)+(3𝐵_[3]⊛^dual_(∗3)(𝑧−𝑧₀))⊛^dual_(∗2)(𝑧−𝑧₀)+⋯}_{𝑖1⋯𝑖𝑞}, (2.41)𝑧∈𝑊₁, uniquely define elements 𝔇_(∗1)𝑇∈𝔗^𝑝(𝑊₁;∗₂,…,∗_𝑀)_(𝑧0) and 𝔇_(∗1)𝑆∈𝔗_𝑞(𝑊₁;∗₂,…,∗_𝑀)_(𝑧0) which will be denoted as the ∗₁-derivative of tensor 𝑇 and the ∗₁-derivative of tensor 𝑆.

We see that the tensors 𝔇_(∗1)𝑇 and 𝔇_(∗1)𝑆 do not contain the operation ∗₁ any longer and so we cannot define 𝔇_(∗1)𝔇_(∗1)𝑇 or 𝔇_(∗1)𝔇_(∗1)𝑆. Only 𝔇_(∗2)𝔇_(∗1)𝑇 and 𝔇_(∗2)𝔇_(∗1)𝑆 make sense, and also 𝔇_(∗3)𝔇_(∗2)𝔇_(∗1)𝑇 and 𝔇_(∗3)𝔇_(∗2)𝔇_(∗1)𝑆, and so forth. Thus we will define the operators

𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}∶=𝔇_(∗𝑘)𝔇_{(∗𝑘−1)}⋯𝔇_(∗1),𝑘∈𝐼_1,𝑀 (2.42) as the chain derivatives of order k. In the special case ∗₁=∗₂=⋯=∗_𝑘∶=∗ we write simplifying 𝔇^𝑘_(∗) instead of 𝔇^𝑘_{(∗,∗,…,∗)}. From the definitions above it follows for 𝑧∈𝑊₁:

(𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑇)^{𝑖1⋯𝑖𝑝}(𝑧)∶={𝐴_[𝑘]+^𝑀_{𝛼≥𝑘+1}𝑃^{[𝛼−𝑘]}_{(∗2,…,∗𝛼)}(𝛼(𝛼−1)⋯(𝛼−𝑘)𝐴_[𝛼],𝑧−𝑧₀)}^{𝑖1⋯𝑖𝑝},(𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑆)_{𝑖1⋯𝑖𝑞}(𝑧)∶={𝐵_[𝑘]+^𝑀_{𝛼≥𝑘+1}𝑃^{dual[𝛼−𝑘]}_{(∗2,…,∗𝛼)}(𝛼(𝛼−1)⋯(𝛼−𝑘)𝐵_[𝛼],𝑧−𝑧₀)}_{𝑖1⋯𝑖𝑞}, (2.43) and we see that

𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑇∈𝔗^𝑝(𝑊₁;∗_𝑘+1,…,∗_𝑀)_(𝑧0),𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑆∈𝔗_𝑞(𝑊₁;∗_𝑘+1,…,∗_𝑀)_(𝑧0). (2.44)

Theorem 2.20. Given a PA1-chain 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) and a chain-analytic tensor𝑇∈𝔗^𝑝(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0),𝑝≥1. Then the following symmetries hold: (𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑇)^{𝑖1⋯𝑖𝑝}_,𝛼=(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑇)^{𝑙𝑖2⋯𝑖𝑝}𝐶^𝑖1_{(∗𝑘+1)𝑙𝛼}+(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑇)^{𝑖1𝑙𝑖3⋯𝑖𝑝}𝐶^𝑖2_{(∗𝑘+1)𝑙𝛼}+⋯+(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑇)^{𝑖1𝑖2⋯𝑖𝑝−1𝑙}𝐶^𝑖𝑝_{(∗𝑘+1)𝑙𝛼}, (2.45)forall𝛼∈𝐼_1,𝑛1,𝑖₁,…,𝑖_𝑝∈𝐼_1,𝑛2,𝑘∈𝐼_0,𝑀−1 (In the case 𝑘=0 the left side must be interpreted as 𝑇^{𝑖1⋯𝑖𝑝}_,𝛼). An inverse relation exists, if the algebra (ℝ^𝑛;∗₁) has a right unit 𝑒_(∗1,𝑅): (𝐷^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑇)^{𝑖1𝑖2𝑖3⋯𝑖𝑝}=1𝑝(𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑇)^{𝑖1⋯𝑖𝑝}_,𝑚𝑒^𝑚_(∗1,𝑅). (2.46)

Theorem 2.21. Given a PA1-chain 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀) and a chain-analytic tensor 𝑆∈𝔗_𝑞(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0),𝑝≥1. Then the following symmetries hold: (𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑆)_{𝑖1⋯𝑖𝑞,𝛼}=(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑆)_{𝑙𝑖2⋯𝑖𝑞}𝐶^𝑙_{(∗𝑘+1)𝑖1𝛼}+(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑆)_{𝑖1𝑙𝑖3⋯𝑖𝑞}𝐶^𝑙_{(∗𝑘+1)𝑖2𝛼}+⋯+(𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑆)_{𝑖1𝑖2⋯𝑖𝑞−1𝑙}𝐶^𝑙_{(∗𝑘+1)𝑖𝑞𝛼}, (2.47)forall𝛼∈𝐼_1,𝑛1,𝑖₁,…,𝑖_𝑝∈𝐼_1,𝑛2,𝑘∈𝐼_1,𝑀−1 (In the case 𝑘=0 the left side must be interpreted as 𝑆_{𝑖1⋯𝑖𝑞,𝛼}). An inverse relation exists, if the algebra (ℝ^𝑛;∗₁) has a right unit 𝑒_(∗1,𝑅): (𝔇^𝑘+1_{(∗𝑘+1,∗𝑘,…,∗1)}𝑆)_{𝑖1𝑖2⋯𝑖𝑞}=1𝑞(𝔇^𝑘_{(∗𝑘,∗𝑘−1,…,∗1)}𝑆)_{𝑖1⋯𝑖𝑞,𝑚}𝑒^𝑚_(∗1,𝑅). (2.48)

The proofs of Theorems 2.20 and 2.21 run the same lines as the proofs of Theorems 2.10 and 2.12. Relations (2.45) and (2.47) will be denoted as Pseudo Cauchy Riemann equations (PCRE) as in the simple case of a single vector space and a single product operation. (The expression𝔇⁰_{(∗0,∗0−1,…,∗1)}𝑇 must be interpreted as 𝑇, and 𝔇⁰_{(∗0,∗0−1,…,∗1)}𝑆 as 𝑆.) Now we will generalize the former concept of ∗-integration.

Definition 2.22. Given an arbitrary algebra 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀)and chain-analytic tensors 𝑇∈𝔗^𝑝(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0),𝑆∈𝔗_𝑞(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0),𝑝≥1, and further let (ℝ^𝑛;∗₀) an algebra with the property (ℝ^𝑛;∗₁,∗₀)isaPA1-structure, (2.49) Then the series (𝔇⁻¹_(∗0)𝑇)^{𝑖1⋯𝑖𝑝}(𝑧)∶={^𝑀_𝛼≥0𝑃^[𝛼+1]_{(∗0,∗1,…,∗𝛼)}(1𝐴𝛼+1^[𝛼],𝑧−𝑧₀)}^{𝑖1⋯𝑖𝑝}+𝐾^{𝑖1⋯𝑖𝑝}∶={𝐾+𝐴_[0]⊛_(∗0)(𝑧−𝑧₀1)+(2𝐴_[1]⊛_(∗1)(𝑧−𝑧₀))⊛_(∗0)(𝑧−𝑧₀)1+((3𝐴_[2]⊛_(∗2)(𝑧−𝑧₀))⊛_(∗1)(𝑧−𝑧₀))⊛_(∗0)(𝑧−𝑧₀)+⋯}^{𝑖1⋯𝑖𝑝},(𝔇⁻¹_(∗0)𝑆)_{𝑖1⋯𝑖𝑞}(𝑧)∶={^𝑀_𝛼≥0𝑃^dual[𝛼+1]_{(∗0,∗1,…,∗𝛼)}(1𝐵𝛼+1^[𝛼],𝑧−𝑧₀)}_{𝑖1⋯𝑖𝑞}+𝐿_{𝑖1…𝑖𝑞}∶={𝐿+𝐵_[0]⊛^dual_(∗0)(𝑧−𝑧₀1)+(2𝐵_[1]⊛^dual_(∗1)(𝑧−𝑧₀))⊛^dual_(∗0)(𝑧−𝑧₀)1+((3𝐵_[2]⊛^dual_(∗2)(𝑧−𝑧₀))⊛^dual_(∗1)(𝑧−𝑧₀))⊛^dual_(∗0)(𝑧−𝑧₀)+⋯}^{𝑖1⋯𝑖𝑝}, (2.50) with 𝑧∈𝑊₁,𝐾 and 𝐿 arbitrary constants, are called the ∗₀-integrals or the ∗₀-antiderivatives of the tensors 𝑇 and 𝑆.

The denotation 𝔇⁻¹_(∗0) is well chosen because it is 𝔇_(∗0)𝔇⁻¹_(∗0)𝑇=𝔇⁻¹_(∗0)𝔇_(∗0)𝑇 and 𝔇_(∗0)𝔇⁻¹_(∗0)𝑆=𝔇⁻¹_(∗0)𝔇_(∗0)𝑆. We see, that

𝑇∈𝔗^𝑝(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0)⇒𝔇⁻¹_(∗0)𝑇∈𝔗^𝑝(𝑊₁;∗₀,∗₁,…,∗_𝑀)_(𝑧0),𝑆∈𝔗_𝑞(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0)⇒𝔇⁻¹_(∗0)𝑆∈𝔗_𝑞(𝑊₁;∗₀,∗₁,…,∗_𝑀)_(𝑧0). (2.51)

Proposition 2.23. If 𝔛is a PA1-chain, then the ∗₀-integrals 𝔇⁻¹_(∗0)𝑇and 𝔇⁻¹_(∗0)𝑆 can be expressed by the classical integral conception as follows: 𝔇⁻¹_(∗0)𝑇(𝑧)=𝑇(𝑧)⊛_(∗0)𝑑𝑧,𝔇⁻¹_(∗0)𝑆(𝑧)=𝑆(𝑧)⊛^dual_(∗0)𝑑𝑧. (2.52)

Theorem 2.24. 𝔛a PA1-chain and 𝔏 a smooth closed curve in region 𝑊₁. Then it is _𝔏𝑇(𝑧)⊛_(∗0)𝑑𝑧=0,_𝔏𝑆(𝑧)⊛^dual_(∗0)𝑑𝑧=0. (2.53)

This Theorem generalizes the well-known classical Cauchy Integral Theorem for GAPs and is denoted as Pseudo Cauchy Integral Theorem (PCIT). For the next step we remember on the classical denotation 𝑦^′=𝑑𝑦/𝑑𝑥 for a scalar differentiable function 𝑦(𝑥), or formally 𝑦^′𝑑𝑥=𝑑𝑦. It shall be shown now, that for the ∗-derivative of vector-valued functions a similar result holds:

Proposition 2.25. Given a PA1-chain 𝔛=(ℝ^𝑛;∗₀,∗₁,…,∗_𝑀)and vectors𝑈∈𝔗¹(𝑊;∗₀,∗₁,…,∗_𝑀)_(𝑧0),𝑉∈𝔗₁(𝑊;∗₀,∗₁,…,∗_𝑀)_(𝑧0). Then it holds 𝑑𝑈(𝑧)=𝔇_(∗0)𝑈⊛_(∗0)𝑑𝑧,𝑑𝑉(𝑧)=𝔇_(∗0)𝑉⊛^dual_(∗0)𝑑𝑧. (2.54)

Proof. We start from the relation (2.52), with 𝑇∈𝔗¹(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0) and 𝑆∈𝔗₁(𝑊₁;∗₁,…,∗_𝑀)_(𝑧0). Now we define 𝑈(𝑧)=𝔇⁻¹_(∗0)𝑇(𝑧),𝑉(𝑧)=𝔇⁻¹_(∗0)𝑆(𝑧), from which follows 𝑈∈𝔗¹(𝑊;∗₀,∗₁,…,∗_𝑀)_(𝑧0),𝑉∈𝔗₁(𝑊;∗₀,∗₁,…,∗_𝑀)_(𝑧0) as it is assumed above. Inserting 𝑈 and 𝑉 in (2.52) we obtain 𝑈(𝑧)=∫𝔇_(∗0)𝑈⊛_(∗0)𝑑𝑧,⇔𝑑𝑈(𝑧)=𝔇_(∗0)𝑈⊛_(∗0)𝑑𝑧, and analog:𝑉(𝑧)=∫𝔇_(∗0)𝑉⊛^dual_(∗0)𝑑𝑧,⇔𝑑𝑉(𝑧)=𝔇_(∗0)𝑉⊛^dual_(∗0)𝑑𝑧.

We sum up in the following paragraph

In this section, the main earlier results have been generalized,along with the algebraic structures as well as the tensor fields built over these structures. In particular the PAk-structures have been generalized by the concept of the so-called “GAPs”. We have investigated special GAPs like PA1 chains and their specializations (closed PA1-chains and closed PA1-chains of Lie-type), and have shown that the main results of the classical complex function theory (Cauchy Riemannian differential equations, Cauchy Integral Theorem) also hold in PA1-chains.

2.4. The Explicit Construction of GAPs

The aim of this section is the explicit construction of a wide variety of GAPs.

In other words, we have to construct structure constant tensors of pseudorings, PAk-structures, and PA1-chains explicitly. This problem makes sense, because without knowledge of concrete GAPs, the results of the sections above would be not applicable in practice. In the following, we introduce the maps 𝐻_𝑛∶ℤ→]0,𝑛], 𝑛∈ℕ (ℤ the set of integers, ℕ the set of natural numbers) with 𝑛≥1 by

𝐻_𝑛(𝑥)=𝑥−𝑚𝑛,if𝑚𝑛<𝑥≤(𝑚+1)𝑛,𝑥if0<𝑥≤𝑛,𝑥+𝑚𝑛if−𝑚𝑛<𝑥≤(−𝑚+1)𝑛, (2.55) where 𝑚∈ℕ denotes the smallest possible value, defined uniquely by the variable 𝑥 and the conditions above (For example, let us consider the case 𝑛=3,𝑥=3.7. Then 𝑚 is uniquely defined by 𝑚=1, since it is 𝑚𝑛<𝑥≤(𝑚+1)𝑛 only for 𝑚=1. Then it is 𝐻_𝑛(𝑥)=𝑥−𝑚𝑛=0.7). From this definition it follows immediately 𝐻_𝑛(𝑥+𝑦)=𝐻_𝑛(𝑦+𝑥), and 𝐻_𝑛(𝑥+𝐻_𝑛(𝑦+𝑧))=𝐻_𝑛(𝑦+𝐻_𝑛(𝑧+𝑥))=𝐻_𝑛(𝑧+𝐻_𝑛(𝑥+𝑦))=𝐻_𝑛(𝑥+𝑦+𝑧).

Proposition 2.26. Given the constants 𝑃∈ℕ,𝑀∈𝐼_1,𝑃,and𝐴^𝑀∈ℝ,𝑎^𝑀,𝜎∈ℤ.Then PAk-structures (ℝ^𝑛;∗₁,∗₂) are given by the following structure constant tensors:case𝑘=1∶𝐶^𝑘_{(∗1)𝑖𝑗}=^𝑃_𝑀=1𝐴^𝑀𝛿^𝑘_{𝐻𝑛(𝑎𝑀𝑖+𝜎𝑗)},𝐶^𝑚_{(∗2)𝑘𝑙}=𝛿^𝑚_{𝐻𝑛(𝑘+𝜎𝑙)},case𝑘=2∶𝐶^𝑘_{(∗1)𝑖𝑗}=𝛿^𝑘_{𝐻𝑛(𝜎𝑖+𝑗)},𝐶^𝑚_{(∗2)𝑘𝑙}=^𝑃_𝑀=1𝐴^𝑀𝛿^𝑚_{𝐻𝑛(𝑎𝑀𝑘+𝜎𝑙)}. (2.56a)(2.56b)

Proof. We show that the algebra (2.56a) satisfies the PA1-condition 𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗2)|𝑘|𝑙]}=0. Writing all sums explicitely we get ∑^𝑛_𝑘=1𝐶^𝑘_{(∗1)𝑖𝑗}𝐶^𝑚_{(∗2)𝑘𝑙}=∑^𝑛_𝑘=1∑^𝑃_𝑀=1𝐴^𝑀𝛿^𝑘_{𝐻𝑛(𝑎𝑀𝑖+𝜎𝑗)}𝛿^𝑚_{𝐻𝑛(𝑘+𝜎𝑙)}=∑^𝑃_𝑀=1𝐴^𝑀𝛿^𝑚_{𝐻𝑛(𝐻𝑛(𝑎𝑀𝑖+𝜎𝑗)+𝜎𝑙)}, because 𝛿^𝑘_{𝐻𝑛(𝑎𝑀𝑖+𝑗)} only delivers a nontrivial value for the index value 𝑘=𝐻^𝑛(𝑎^𝑀𝑖+𝑗) and this index 𝑘 must be inserted in the term 𝛿^𝑚_{𝐻𝑛(𝑘+𝜎𝑙)}. Since it is 𝐻_𝑛(𝐻_𝑛(𝑎^𝑀𝑖+𝜎𝑗)+𝜎𝑙)=𝐻_𝑛(𝑎^𝑀𝑖+𝜎𝑗+𝜎𝑙), the right side above is symmetrical in indices 𝑗 and 𝑙. Thus the left side is symmetrical in indices 𝑗 and 𝑙, that is, the PA1-symmetry holds and statement (2.56a) has been proved. Now we have to show that algebra (2.56b) satisfies the PA2-condition 𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗1)𝑙]𝑘} =0∶∑^𝑛_𝑘=1𝐶^𝑘_{(∗2)𝑖𝑗}𝐶^𝑚_{(∗1)𝑙𝑘} =∑^𝑛_𝑘=1∑^𝑃_𝑀=1𝐴^𝑀𝛿^𝑘_{𝐻𝑛(𝑎𝑀𝑖+𝜎𝑗)}𝛿^𝑚_𝐻^𝑛_{(𝜎𝑙+𝑘)} = ∑^𝑃_𝑀=1𝐴^𝑀𝛿^𝑚_{𝐻𝑛(𝜎𝑙+𝐻𝑛(𝑎𝑀𝑖+𝜎𝑗))} = 𝛿^𝑚_{𝐻𝑛(𝜎𝑙+𝑎𝑀𝑖+𝜎𝑗)}. The right side is symmetrical in indices 𝑗 and 𝑙, and so the left side is, that is, the PA2-symmetry holds and statement (2.56b) has been proved.

Proposition 2.27. Let be 𝑛∈ℕ,𝜎∈ℤarbitrary. Then the structure constant tensor 𝐶^𝑘_{(∗)𝑖𝑗}=𝛿^𝑘_{𝐻𝑛(𝑖+𝜎𝑗)},𝑖,𝑗,𝑘∈𝐼_1,𝑛 (2.57) defines a pseudoring of first type (ℝ^𝑛;∗). In the case 𝜎=1, a unit element exists, given by 𝑒^𝑗_(∗)=𝛿^𝑗_𝑛, and furthermore it hold s that∃𝜆∈ℝ^𝑛,𝜆≠0with∄𝜆^−1(∗),Anonexisting𝜆^−1(∗)doesnotimply𝜆²=0. (2.58a)(2.58b)

Because the PAk-structures above are defined by Kronecker-symbols, we will denote them as PAk-structures of Kronecker-type or shorter as Kronecker PAk-structures. According to the property (2.58a) a commutative Kronecker pseudoring is not a division algebra. It is easy to show, that this property also holds for noncommutative Kronecker pseudorings. Also the property (2.58a) can be generalized for noncommutative Kronecker pseudorings, where 𝜆^−1(∗) must be replaced by 𝜆^{−1(∗,𝑅,𝑅)}, and so forth. (see the definition of 𝜆^{−1(∗,𝑅,𝑅)},𝜆^{−1(∗,𝑅,𝐿)},𝜆^{−1(∗,𝐿,𝑅)},𝜆^{−1(∗,𝐿,𝐿)} in (2.2)). We see that a Kronecker algebra has a very simple structure, because the values of the structure constants are given by only two reals: 0 and 1. Now another type of PAk-structures will be presented which is more subtle than the Kronecker type.

Proposition 2.28. Given an algebra (ℝ^𝑛;∗₁,∗₂) as follows: 𝐶^ℎ_{(∗𝛼)𝑖𝑗}=𝑑_𝑖𝛿^ℎ_𝑗+𝑏_(∗𝛼)𝑑_𝑗𝛿^ℎ_𝑖+𝑑_𝑖𝑑_𝑗ℎ^ℎ_(∗𝛼),𝛼=1,2,(informaldenotation∶𝐶_(∗𝛼)=𝑑⊗𝐼+𝑏_(∗𝛼)𝐼⊗𝑑+𝑑⊗𝑑⊗ℎ_(∗𝛼)) (2.59) with 𝑑,ℎ_(∗1),ℎ_(∗2)∈ℝ^𝑛, 𝑑≠0,𝑏_(∗1),𝑏_(∗2)∈ℝ arbitrary nonzero constants. Then (ℝ^𝑛;∗₁,∗₂) is a PA𝛼-structure if and only if ⟨𝑑,ℎ_(∗𝛼)⟩=−1+𝜎(𝛼){𝑏_(∗2)−𝑏^𝜎(𝛼)_(∗1)}with𝜎(𝛼)=(−1)^𝛼−1,𝛼=1,2. (2.60)(ℝ^𝑛;∗_𝛼),𝛼=1,2has a right unit 𝑒_{(∗𝛼,𝑅)} if and only if ⟨𝑑,ℎ_(∗𝛼)⟩=−1,Thenitis𝑒_{(∗𝛼,𝑅)}=−𝑏⁻¹_(∗𝛼)ℎ_(∗𝛼). (2.61)

The PAk-structures presented above are built by combinations of rank 2-tensors and rank 1-tensors. Therefore we will denote these structures as PAk-structures of splitting type or shorter as splitting PAk-structures. From this follows immediately the structure of splitting pseudorings. It is easy to show, that Kronecker PAk-structures generally cannot be transformed onto splitting PAk-structures by basistransformations in ℝ^𝑛, that is, these both structures are really different.

Splitting pseudorings have a large advantage to Kronecker pseudorings because it allow the explicit calculation of √^𝑃𝑧,𝑧⁻¹, and so forth.

Proposition 2.29. Given the structure constant tensor 𝐶^𝑘_{(∗)𝑖𝑗}=𝑑_𝑖𝛿^𝑘_𝑗+𝑏𝑑_𝑗𝛿^𝑘_𝑖+𝑑_𝑖𝑑_𝑗ℎ^𝑘withℎ^𝑘𝑑_𝑘=−1,(informaldenotation∶𝐶_(∗𝛼)=𝑑⊗𝐼+𝑏𝐼⊗𝑑+𝑑⊗𝑑⊗ℎ,with⟨ℎ,𝑑⟩=−1) (2.62)𝑖,𝑗,𝑘∈𝐼_1,𝑛,𝑏∈ℝ only restricted by 𝑏≠0. Then the following statements hold:

Statement 1. The algebra(ℝ^𝑛;∗)defines an 𝑛-dimensional pseudoring of first type with right unit 𝑒_(∗,𝑅)=−𝑏⁻¹ℎ.

Statement 2. In this algebra a vector 𝑧 has a right invers element 𝑧^{−1(∗,𝑅,𝑅)} in respect of the right unit, if and only if ⟨𝑧,𝑑⟩≠0.

Statement 3. In (ℝ^𝑛;∗) the power functions, roots and inverses of vectors are calculable explicitly as follows:𝑧^[𝑃]=𝑏^𝑃−2{(𝑏+𝑃−1)⟨𝑑,𝑧⟩^𝑃−1𝑧+(𝑃−1)⟨𝑑,𝑧⟩^𝑃𝑧ℎ},𝑃∈ℕ,_{−1(∗,𝑅,𝑅)}=−𝑏⁻²{𝑏⟨𝑑,𝑧⟩⁻²𝑧+(1+𝑏)⟨𝑑,𝑧⟩⁻¹ℎ}, (2.63)(2.64) where 𝑧^[𝑃] is defined by 𝑧^[𝑃]=(⋯((𝑧∗𝑧)∗𝑧)⋯)∗𝑧. In the case 𝑏=1 the relation (2.63) also holds for arbitrary rational numbers.

Statement 4. For 𝑏=1 it holds𝐴nonexisting𝜆^−1(∗)implies𝜆²=0,𝑈∈𝔗¹(𝑊;∗)_(𝑧0)⇒𝜆<𝑑,𝑈>∈𝔗¹(𝑊;∗)_(𝑧0)iff∄𝜆^−1(∗),𝑈∈𝔗₁(𝑊;∗)_(𝑧0)⇒𝑑<𝜆,𝑈>∈𝔗₁(𝑊;∗)_(𝑧0)iff∄𝜆^−1(∗). (2.65)

Proposition 2.29 shows, that in the framework of splitting pseudorings the inverse elements as well as the roots of vectors are calculable explicitly, that is, without using numerical methods. In Kronecker algebras this is not possible. Now we will show, that PA1-chains and furthermore PA1-chains of Lie-type can be constructed explicitly, proving, that the world of the PAk-structures is compatible with the world of generalized Lie-symmetries:

Proposition 2.30. The following algebraic structures define PA1-chains 𝔛=(ℝ^𝑛;∗₁,∗₂,…,∗_𝑀)∶Kronecker PA1-chains:𝐶^𝑘_{(∗𝛼)𝑖𝑗}=𝛿^𝑘_{𝐻𝑛(𝑖+𝜎𝑗)},𝛼=1,…,𝑀,𝜎∈ℤarbitrary, (2.66a) Splitting PA1-chains: 𝐶^𝑘_{(∗𝛼)𝑖𝑗}=𝑑_𝑖𝛿^𝑘_𝑗+𝑏_(∗𝛼)𝑑_𝑗𝛿^𝑘_𝑖+𝑑_𝑖𝑑_𝑗ℎ^𝑘_(∗𝛼),𝛼∈𝐼_1,𝑀,with1+𝑏_(∗𝛼)+⟨𝑑,ℎ_(∗𝛼)⟩=𝑏_{(∗𝛼−1)},𝛼∈𝐼_2,𝑀. (2.66b) Modified Splitting 𝐶^𝑘_{(∗𝛼)𝑖𝑗}=𝑑_{(∗𝛼)𝑖}𝐴^𝑘_{(∗𝛼)𝑗}+𝑔_𝑗𝐵^𝑘_{(∗𝛼)𝑖},𝛼∈𝐼_1,𝑀,withtherestrictions (2.66c) PA1-chains: 𝐴^𝑘_{(∗𝛼+1)𝑗}𝑑_{(∗𝛼)𝑘}=0,𝐵^𝑘_{(∗𝛼+1)𝑗}𝑑_{(∗𝛼)𝑘}=𝑢𝑑_{(∗𝛼+1)𝑗},𝐴^𝑘_{(∗𝛼+1)𝑗}𝐵^𝑚_{(∗𝛼)𝑘}=𝑢𝐴^𝑚_{(∗𝛼)𝑗},𝑢∈ℝarbitrary. (2.66d)

Proposition 2.31. The following algebraic structure (ℝ^𝑛;∗₁,∗₂,∗₃) defines a closed PA1-chain from Lie type: 𝐶^𝑘_{(∗𝛼)𝑖𝑗}=𝑑_𝑖𝛿^𝑘_𝑗+𝑏_(∗𝛼)𝑑_𝑗𝛿^𝑘_𝑖+𝑑_𝑖𝑑_𝑗ℎ^𝑘_(∗𝛼),𝛼∈𝐼_1,3𝑏,with_(∗1)=13(⟨ℎ_(∗2),𝑑⟩−⟨ℎ_(∗1)𝑏,𝑑⟩),_(∗2)2=−1+3⟨ℎ_(∗2)1,𝑑⟩+3⟨ℎ_(∗1)𝑏,𝑑⟩,_(∗3)=1+⟨ℎ_(∗1)ℎ,𝑑⟩,_𝑚(∗1)𝑏_(∗2)+ℎ^𝑚_(∗2)𝑏_(∗3)+ℎ^𝑚_(∗3)𝑏_(∗1)=0. (2.67)

Until now we have delt with special PAk -structures (Kronecker structures and splitting structures). Now we will analyze the problem, if PAk-structures can be combined in a way, that we come to PAk-structures again. There are two ways to analyze this problem. The first way is to try a construction of new 𝑛-dimensional PAk-structure by combining 𝑛-dimensional PAk-structures, the second way is to try a construction of new 𝑁-dimensional PAk-structure by combining 𝑛-dimensional PAk-structures, 𝑁>𝑛. We will give answers to both problems by the following both propositions.

Proposition 2.32. Given a GAP 𝔛=(ℝ^𝑛;∗₁,∗₂,∗₃,∗₄)with the characteristic 𝐺_(𝔛)=0010000100000000. (2.68) Then the algebra 𝔜=(ℝ^𝑛;⊛₁,⊛₂)defined by 𝐶^𝑘_{(⊛1)𝑖𝑗}∶=𝐶^𝑘_{(∗1)𝑖𝑗}+𝐶^𝑘_{(∗2)𝑖𝑗},𝐶^𝑘_{(⊛2)𝑖𝑗}∶=𝐶^𝑘_{(∗3)𝑖𝑗}+𝐶^𝑘_{(∗4)𝑖𝑗} (2.69) is a PA1-structure, if and only if 𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗3)|𝑘|𝑙]}=−𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗4)|𝑘|𝑙]}. (2.70) Given a GAP 𝔛=(ℝ^𝑛;∗₁,∗₂,∗₃,∗₄)with the characteristic: 𝐺_(𝔛)=0100001000000000. (2.71) Then the algebra ℨ=(ℝ^𝑛;⊚₁,⊚₂)defined by 𝐶^𝑘_{(⊚1)𝑖𝑗}∶=𝐶^𝑠_{(∗1)𝑖𝑗}𝐶^𝑘_{(∗2)𝑠𝑝}𝑢^𝑝,𝐶^𝑘_{(⊚2)𝑖𝑗}∶=𝐶^𝑠_{(∗3)𝑖𝑗}𝐶^𝑘_{(∗4)𝑠𝑝}𝑣^𝑝,𝑢,𝑣∈ℝ^𝑛arbitrary (2.72) is a PA1-structure.

Proof. To prove the first statement we calculate 𝐶^𝑘_{(⊛1)𝑖𝑗}𝐶^𝑚_{(⊛2)𝑘𝑙}=(𝐶^𝑘_{(∗1)𝑖𝑗}+𝐶^𝑘_{(∗2)𝑖𝑗})(𝐶^𝑚_{(∗3)𝑘𝑙}+𝐶^𝑚_{(∗4)𝑘𝑙})=𝐶^𝑘_{(∗1)𝑖𝑗}𝐶^𝑚_{(∗3)𝑘𝑙}+𝐶^𝑘_{(∗1)𝑖𝑗}𝐶^𝑚_{(∗4)𝑘𝑙}+𝐶^𝑘_{(∗2)𝑖𝑗}𝐶^𝑚_{(∗3)𝑘𝑙}+𝐶^𝑘_{(∗2)𝑖𝑗}𝐶^𝑚_{(∗4)𝑘𝑙},⇒𝐶^𝑘_{(⊛1)𝑖[𝑗}𝐶^𝑚_{(⊛2)|𝑘|𝑙]}=𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗3)|𝑘|𝑙]}+𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗4)|𝑘|𝑙]}+𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗3)|𝑘|𝑙]}+𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗4)|𝑘|𝑙]GAP-symmetries(2.69)}=𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗4)|𝑘|𝑙]}+𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗3)|𝑘|𝑙]}. (2.73) We see, that the PA1 -symmetry 𝐶^𝑘_{(⊛1)𝑖[𝑗}𝐶^𝑚_{(⊛2)|𝑘|𝑙]}=0 holds if and only if 𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗4)|𝑘|𝑙]}+𝐶^𝑘_{(∗2)𝑖[𝑗}𝐶^𝑚_{(∗3)|𝑘|𝑙]}=0, according to (2.70). To prove the second statement we calculate 𝐶^𝑘_{(⊚1)𝑖𝑗}𝐶^𝑚_{(⊚2)𝑘𝑙}=𝐶^𝑠_{(∗1)𝑖𝑗}𝐶^𝑘_{(∗2)𝑠𝑝}𝑢^𝑝𝐶^𝑟_{(∗3)𝑘𝑙}𝐶^𝑚_{(∗4)𝑟𝑞}𝑣^𝑞,⇒𝐶^𝑘_{(⊚1)𝑖[𝑗}𝐶^𝑚_{(⊚2)|𝑘|𝑙]}=𝐶^𝑠_{(∗1)𝑖[𝑗}𝐶^𝑘_{(∗2)|𝑠𝑝}𝑢^𝑝𝐶^𝑟_{(∗3)𝑘|𝑙]}𝐶^𝑚_{(∗4)𝑟𝑞}𝑣^𝑞_{GAP-symmetries(2.72)}=𝐶^𝑠_{(∗1)𝑖[𝑗}𝐶^𝑘_{(∗2)|𝑠|𝑙]}𝑢^𝑝𝐶^𝑟_{(∗3)𝑘𝑝}𝐶^𝑚_{(∗4)𝑟𝑞}𝑣=^{𝑞GAP-symmetries(2.72)}0. (2.74)

Proposition 2.33. Given a PA1-structure𝔛=(ℝ^𝑛;∗₁,∗₂). Then the following algebra 𝔜=(ℝ^2𝑛;⊛) is a pseudoring of first type: 𝐶^𝑘_{(⊛)𝑖𝑗}=𝐶^𝑘_{(∗1)𝑖𝑗},if𝑖,𝑗,𝑘∈𝐼_1,𝑛,𝐶^𝑘−𝑛_{(∗2)𝑖−𝑛𝑗−𝑛},if𝑖,𝑗,𝑘∈𝐼_𝑛+1,2𝑛,0,inallothercases. (2.75)

Proof. Per defnition it is 𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=^𝑛_𝑘=1𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}+^2𝑛_𝑘=𝑛+1𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙} (2.76) for arbitrary indices 𝑖,𝑗,𝑙,𝑚∈𝐼_1,2𝑛. Let us consider first the case that 𝑖∈𝐼_1,𝑛. Then nontrivial elements of 𝐶^𝑘_{(⊛)𝑖𝑗} only exist in the case 𝑗,𝑘∈𝐼_1,𝑛, defined by 𝐶^𝑘_{(⊛)𝑖𝑗}=𝐶^𝑘_{(∗1)𝑖𝑗}, and the term above takes the form 𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=^𝑛_𝑘=1𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=^𝑛_𝑘=1𝐶^𝑘_{(∗1)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=0,if𝑙>𝑛or𝑚>𝑛,_𝑛𝑘=1𝐶^𝑘_{(∗1)𝑖𝑗}𝐶^𝑚_{(∗2)𝑘𝑙},if𝑙,𝑚∈𝐼_1,𝑛. (2.77) Thus it follows for 𝑖∈𝐼_1,𝑛∶𝐶^𝑘_{(⊛)𝑖[𝑗}𝐶^𝑚_{(⊛)|𝑘|𝑙]}=0,if𝑙>𝑛or𝑚>𝑛_𝑛𝑘=1𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗2)|𝑘|𝑙]},if𝑙,𝑚∈𝐼_1,𝑛=0 (2.78) in every case, because the algebra (ℝ^𝑛;∗₁,∗₂) is assumed as a PA1-structure, that is, it is ∑^𝑛_𝑘=1𝐶^𝑘_{(∗1)𝑖[𝑗}𝐶^𝑚_{(∗2)|𝑘|𝑙]}=0 for 𝑖∈𝐼_1,𝑛.
Let us consider now the case that 𝑖∈𝐼_𝑛+1,2𝑛. Then nontrivial elements of 𝐶^𝑘_{(⊛)𝑖𝑗} only exist in the case 𝑗,𝑘∈𝐼_𝑛+1,2𝑛, defined by 𝐶^𝑘_{(⊛)𝑖𝑗}=𝐶^𝑘−𝑛_{(∗2)𝑖−𝑛𝑗−𝑛}, and we get
𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=^2𝑛_𝑘=𝑛+1𝐶^𝑘_{(⊛)𝑖𝑗}𝐶^𝑚_{(⊛)𝑘𝑙}=^2𝑛_𝑘=𝑛+1𝐶^𝑘−𝑛_{(∗1)𝑖−𝑛𝑗−𝑛}𝐶^𝑚_{(⊛)𝑘𝑙}=0,if𝑙≤𝑛or𝑚≤𝑛,_{2𝑛𝑘=𝑛+1}𝐶^𝑘−𝑛_{(∗1)𝑖−𝑛𝑗−𝑛}𝐶^𝑚−𝑛_{(∗2)𝑘−𝑛𝑙−𝑛},if𝑙,𝑚∈𝐼_𝑛+1,2𝑛. (2.79) Thus it follows with the denotations 𝜄=𝑖−𝑛,𝑗=𝑗−𝑛,𝑘=𝑘−𝑛,𝑙=𝑙−𝑛,𝑚=𝑚−𝑛∶𝐶^𝑘_{(⊛)𝑖[𝑗}𝐶^𝑚_{(⊛)|𝑘|𝑙]}=0if𝑙≤𝑛or𝑚≤𝑛,_𝑛𝑘=1𝐶^𝑘_{(∗1)𝜄[𝑗}𝐶^𝑚_{(∗2)|𝑘|𝑙]}inallothercases,=0. (2.80) in every case, because the indices 𝜄,𝑗,𝑘,𝑙,𝑚 move in the region 𝐼_1,𝑛, and the algebra (ℝ^𝑛;∗₁,∗₂) is assumed as a PA1-structure.

In this section a wide variety of GAPs has been constructed explicitly. In particular two different types of explicit calculable PAk-structures have been characterized: Kronecker PAk-structures and (generalized) Splitting PAk-structures. Also PA1-chains and even PA1-chains of Lie-type can be constructed explicitly.

3. Applications of GAP-Theory on Solving Partial Differential Equations. New Explicit Solutions of Einstein Equations and Navier-Stokes Equations

In this chapter we will show, that GAP-Theory allows interesting applications on Theoretical Physics by calculating new explicit solutions of Einstein's field equations from General Relativity Theory and new explicit solutions of Navier-Stokes equations.

In some cases, the gained solutions are presented not only in the (compact and elegant) notation of GAP-Theory but also in the more circumstantial classical notation. This allows the checking of the results by readers without knowledges in GAP-Theory (Assumed are only the definitions of the vector spaces 𝔗¹(𝑊;∗)_(𝜉0) and 𝔗₁(𝑊;∗)_(𝜉0) from Section 2.1.) by inserting the presented solutions into the corresponding differential equations.

3.1. New Explicit Solutions of Einstein Equations

We start with Einsteins field equations 𝑅_𝑖𝑗−12𝑅𝑔_𝑖𝑗=−𝜅𝑇_𝑖𝑗. (3.1) Here 𝑔_𝑖𝑗 denotes the metric tensor of an 𝑛-dimensional Riemannian manifold 𝔐 with relativistic signature, 𝑅_𝑖𝑗 the Ricci tensor defined by 𝑅_𝑖𝑗=𝑔^𝑚𝑘𝑅_{𝑚𝑖𝑗𝑘}=12𝑔^𝑚𝑘(𝑔_{𝑚𝑘,𝑖𝑗}+𝑔_{𝑖𝑗,𝑚𝑘}−𝑔_{𝑖𝑘,𝑗𝑚}−𝑔_{𝑗𝑚,𝑖𝑘})+𝑔^𝑚𝑘𝑔^ℎ𝑙(Γ_ℎ𝑖𝑗Γ_𝑙𝑚𝑘−Γ_ℎ𝑖𝑘Γ_𝑙𝑚𝑗Γ),_ℎ𝑖𝑗=12(𝑔_ℎ𝑖,𝑗−𝑔_𝑖𝑗,ℎ+𝑔_𝑗ℎ,𝑖), (3.2)Γ_ℎ𝑖𝑗 the Christoffel symbols of first kind, and 𝑅∶=𝑔^𝑖𝑗𝑅_𝑖𝑗 the curvature scalar. The right side of Einstein equations is defined by the relativistic gravitational constant 𝜘=8𝜋𝛾/𝑐⁴=1.86⋅10⁻²⁷ cm/g (𝛾 the classical gravitational constant) and the energy momentum tensor 𝑇_𝑖𝑗 of the considered matter.

Einstein equations represent a system of nonlinear PDEs of second order to calculate the covariant components 𝑔_𝑖𝑗 of the metric. The left side of these equations is linear in the second derivatives but nonlinear in the first derivatives of the covariant metric 𝑔_𝑖𝑗 and also nonlinear in the covariant metric itself, because 𝑔^𝑚𝑘 is a nonlinear function of 𝑔_𝑖𝑗. Generally also the right side of Einstein equations is nonlinear in the metric and the type of nonlinearity depends on the concrete physically situation.

In the past a lot of explicit solutions of Einstein equations have been constructed, but most of them are restricted on the very special case 𝑅_𝑖𝑗=𝜆𝑔_𝑖𝑗 (Einstein spaces). Among these solutions only a small set is of physical importance because most of them have too special mathematical symmetries for describing physically gravitational fields. By contrast we will develope a variety of explicit solutions in the framework of GAP-Theory without assuming Einstein spaces.

Algebraic tools for solving Einstein equations have been used in [18, 19] (associative and commutative algebras) and [4] (nonassociative and noncommutative algebras; see also the overview given in [4]), discussions of certain generalizations of Einstein equations with algebra-valued metrics. Unlike we will deal here with the classical Einstein equations, that is, with real-valued, four dimensional metrics, and special GAPs are used for describing certain inherent symmetries of the explicit solutions. These GAPs differ from the algebras used in the papers cited above.

Finally some remarks to the denotions in this chapter are presented: 𝜂_𝑖𝑗 is the covariante Minkowski metric, 𝜂^𝑗𝑘 defined by 𝜂_𝑖𝑗𝜂^𝑗𝑘=𝛿^𝑘_𝑗. In relativistic theories, the indices run from 0 to 3 or—for generalizations—from 0 to 𝑛. In the framework of algebra theory, however, the left index range is not 0 but 1 and the relativistic denotion would lead to some problems and misunderstandings. For this we always will use the index range 1,…,4 or—for generalizations—from 1 to 𝑛. Furthermore, sometimes we will write ⟨𝑢,𝑣⟩ for 𝑔_𝑖𝑗𝑢^𝑖𝑣^𝑗.

Theorem 3.1 (in GAP-notation). Given a four-dimensional splitting pseudoring (𝑅⁴;∗) with the splitting parameters 𝑏,𝑑,ℎ and the restrictions 𝑏=1or𝑏=−3,𝑑_𝑖𝑑_𝑗𝜂^𝑖𝑗=0, (3.3) and further given an arbitrary vector field 𝑎∈𝔗¹(𝑊;∗)_(𝜉0),  𝑊⊆𝔐, then all metric tensors of the form 𝑔_𝑖𝑗(𝜉)=𝑎^𝑚(𝜉)𝐶^𝑙_{(∗)𝑚{𝑖}𝜂_𝑗}𝑙 (3.4) (𝐶_(∗) the corresponding structure constant tensor, 𝐶^𝑙_{(∗)𝑚{𝑖}𝜂_𝑗}𝑙∶=𝐶^𝑙_{(∗)𝑚𝑖}𝜂_𝑗𝑙+𝐶^𝑙_{(∗)𝑚𝑗}𝜂_𝑖𝑙) are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system (𝜉).

Theorem 3.2 (in classical notation). Given the reals 𝑏,𝑑_𝑖,ℎ^𝑖,  𝑖=1,…,4 with the restrictions 𝑏=1or𝑏=−3,d_id_j𝜂^ij=0,d_ihⁱ=−1, (3.5)
And further given an arbitrary vector field 𝑎∈𝔗¹(𝑊;∗)_(𝜉0),  𝑊⊆𝔐, then all metric tensors of the form 𝑔_𝑖𝑗(𝜉)=2𝑎^𝑚(𝜉)𝑑_𝑚𝜂_𝑖𝑗+𝑏[𝑑_𝑖𝑎^𝑚(𝜉)𝜂_𝑚𝑗+𝑑_𝑗𝑎^𝑚(𝜉)𝜂_𝑚𝑖]+(𝑎^𝑚(𝜉)𝑑_𝑚)[𝑑_𝑖ℎ^𝑙𝜂_𝑙𝑗+𝑑_𝑗ℎ^𝑙𝜂_𝑙𝑖] (3.6) are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system ( 𝜉).

The proof is omitted for space reason, the interested reader may insert the solution into the vacuum Einstein equations 𝑅_𝑖𝑗=0. It is remarkable that vacuum solutions are realizable in the case of commutative algebras (𝑏=1) as well as in the case of noncommutative algebras (𝑏=−3).

Now we will show that the vacuum metric above is a new unknown metric and not only a certain representation of the well-known Kerr metric or if the metric in [17]. For this we remember on the structure of the Kerr metric, which is given in Boyer-Lindquist coordinates 𝑡,𝑟,𝜙,𝜃 as follows: 𝑑𝑠²𝑟=(1−^𝑠𝑟𝜌²)𝑐²𝑑𝑡²+2𝑟_𝑠𝑟𝑎sin²𝜃𝜌²𝜌𝑐𝑑𝑡𝑑𝜙−²Λ²𝑑𝑟²−𝜌²𝑑𝜃²−(𝑟²+𝑎²+𝑟_𝑠𝑟𝑎²𝜌²sin²𝜃)𝑑𝜙², (3.7)

𝑟_𝑠 the Schwarzschild radius, 𝑎 the Kerr parameters and 𝜌²∶=𝑟²+𝑎²cos²𝜃,  Λ²∶=𝑟²−𝑟_𝑠𝑟+𝑎². In the following matrix form: (𝑔_𝑖𝑗𝑟)=(1−^𝑠𝑟𝜌²)02𝑟_𝑠𝑟𝑎sin²𝜃𝜌²𝜌𝑐0⋯−²Λ²00⋯⋯−(𝑟²+𝑎²+𝑟_𝑠𝑟𝑎²𝜌²sin²𝜃)0⋯⋯⋯−𝜌², (3.8) remember that the two fundamental symmetries of the Kerr metric—which describes an uncharged rotating black hole—are given by (𝜕/𝜕𝑡)𝑔_𝑖𝑗=0 (time-independence) and (𝜕/𝜕𝜙)𝑔_𝑖𝑗=0 (axial symmetry).

The matrix form of our metric (3.6) is given by (𝑔_𝑖𝑗)=2(𝜓+𝑑₁𝑤₁)𝑑₁𝑤₂+𝑑₂𝑤₁𝑑₁𝑤₃+𝑑₃𝑤₁𝑑₁𝑤₄+𝑑₄𝑤₁⋯2(−𝜓+𝑑₂𝑤₂)𝑑₂𝑤₃+𝑑₃𝑤₂𝑑₂𝑤₄+𝑑₄𝑤₂⋯⋯2(−𝜓+𝑑₃𝑤₃)𝑑₃𝑤₄+𝑑₄𝑤₃⋯⋯⋯2(−𝜓+𝑑₄𝑤₄), (3.9) with 𝑤_𝑗(𝜉)∶=𝜂_𝑗𝑘(𝑏𝑎^𝑘+ℎ^𝑘),𝜓(𝜉)∶=𝑎^𝑘𝑑_𝑘,𝑎∈𝔗¹(𝑊;∗)_(𝜉0) an arbitrary field. It is easy to see that there does not exist any coordinate transformation such that the metric (3.9) takes the structure of the Kerr metric (3.8). The zeros in (3.8) cannot be realized by preserving the structure of the nontrivial elements, that is, our vacuum metric is not from Kerr type; the Kerr symmetries of time-independence and axial symmetry are replaced by the symmetry of ∗-analyticity.

It remains to show that the metric is not from the type presented in [17]. For this we compare the following: Metricin[17]∶𝑔_𝑖𝑗(𝜉)=𝜂_𝑖𝑗+𝐶^𝑙_{(∗){𝑖𝑗}}𝐴_𝑙(𝜉)=𝜂_𝑖𝑗+(𝐶^𝑙_{(∗)𝑖𝑗}+𝐶^𝑙_{(∗)𝑗𝑖})𝐴_𝑙(𝜉),Metricabove∶𝑔_𝑖𝑗(𝜉)=𝑎^𝑚(𝜉)(𝐶^𝑙_{(∗)𝑚{𝑖}𝜂_𝑗}𝑙=𝑎^𝑚(𝜉)(𝐶^𝑙_{(∗)𝑚𝑖}𝜂_𝑗𝑙+𝐶^𝑙_{(∗)𝑚𝑗}𝜂_𝑖𝑙) (3.10) with 𝐴∈𝔗₁(𝑊;∗)_(𝜉0),  𝑎∈𝔗¹(𝑊;∗)_(𝜉0). It is easy to see that the metrics are of different types.

(i)The intrinsic symmetries of the vector fields 𝐴∈𝔗₁(𝑊;∗)_(𝜉0) and 𝑎∈𝔗¹(𝑊;∗)_(𝜉0)  are of different types (𝔗₁(𝑊;∗)_(𝜉0)≠𝔗¹(𝑊;∗)_(𝜉0)).(ii)The formal structure of the metrics differs, because the structure constant tensor appears in different manners: (𝐶^𝑙_{(∗)𝑖𝑗}𝐴_𝑙(𝜉)≠𝑎^𝑚(𝜉)𝐶^𝑙_{(∗)𝑚𝑖},𝐶^𝑙_{(∗){𝑖𝑗}}≠𝐶^𝑙_{(∗)𝑚{𝑖}𝜂_𝑗}𝑙).(iii)The metric in [17] cannot describe nontrivial vacuum solutions (i.e., vacuum solutions for nonflat spaces; see [17]), whereas our new metric describes nontrivial vacuum solutions.

For these reasons, we recognize the metric presented here as a new, unknown vacuum metric. Now we will solve Einstein equations for a nonvanishing stress energy tensor.

Theorem 3.3 (in GAP-notation). Given an 𝑛-dimensional splitting pseudoring (ℝ^𝑛;∗) with the splitting parameters 𝑏,𝑑,ℎand the restrictions 𝑑_𝑖𝑑_𝑗𝜂^𝑖𝑗=0, (3.11) and further given the 𝑛-dimensional Einstein equations in a local coordinate system (𝜉) in the form 𝑅_𝑖𝑗−12𝑅𝑔_𝑖𝑗=−𝜘𝜀𝑑_𝑖𝑑_𝑗with𝜀(𝜉)=−𝜘⁻¹𝑏²(𝑛−2)⟨𝑤,𝔇²_(∗)𝑓⟩−𝑏⟨𝑤,𝔇_(∗)𝑓⟩⟨𝑑,𝔇_(∗)𝑎⟩2(1+⟨𝑤,𝔇_(∗)𝑓⟩),𝑤^𝑙𝑑_𝑙=0, (3.12)𝑎,𝑓∈𝔗₁(𝑊;∗)_(𝜉0), 𝑊⊆𝔐arbitrary vector fields, restricted only by the condition 𝑎_𝑖≠𝛾𝑑_𝑖,  𝛾 a scalar field, 𝑤^𝑙constants, then these equations can be solved explicitly by 𝑔_𝑖𝑗(𝜉)=(1+⟨𝑤,𝔇_(∗)𝑓⟩)𝜂_𝑖𝑗+(𝐶^ℎ_{(∗)𝑖𝑗}+𝐶^ℎ_{(∗)𝑗𝑖})𝑎_ℎ(𝜉), (3.13)𝐶_(∗)the structure constant tensor of the splitting pseudoring.

Theorem 3.4 (in classical notation). Given the reals 𝑏,𝑑_𝑖,ℎ^𝑖,𝑤^𝑖,𝑖=1,…,𝑛with the restrictions 𝑑_𝑖𝑑_𝑗𝜂^𝑖𝑗=0,ℎ^𝑖𝑑_𝑖=−1,𝑤^𝑙𝑑_𝑙=0, (3.14) and further given the n-dimensional Einstein equations in a local coordinate system (𝜉) in the form 𝑅_𝑖𝑗−12𝑅𝑔_𝑖𝑗=−𝜘𝜀(𝜉)𝑑_𝑖𝑑_𝑗, (3.15)𝜀a scalar field, these equations can be solved explicitly by 𝑔_𝑖𝑗(𝜉)=(1+𝑤^𝑚(𝔇_(∗)𝑓)_𝑚)𝜂_𝑖𝑗+(𝑏+1)[𝑑_𝑖𝑎_𝑗(𝜉)+𝑑_𝑗𝑎_𝑖(𝜉)]+2𝑑_𝑖𝑑_𝑗ℎ^𝑚𝑎_𝑚(𝜉), (3.16) where 𝐶_(∗)𝑖𝑠the structure constant tensor of the splitting pseudoring, 𝑎,𝑓∈𝔗₁(𝑊;∗)_(𝜉0),  𝑊⊆𝔐arbitrary vector fields, restricted only by the condition 𝑎_𝑖≠𝛾𝑑_𝑖,  𝛾a scalar field. The concrete structure of the vector fields 𝑎,𝑓defines the structure of the scalar field 𝜀.

The proof is omitted for space reason, the interested reader may insert the solution above into the Einstein equations. Theorem 3.4 is less general than Theorem 3.3 because nothing is told about the concrete structure of the scalar field 𝜀. (The classical notation of 𝜀 would be very circumstantial.) We see that solutions of the inhomogeneous 𝑛-dimensional Einstein equations are realizable in the case of commutative algebras (𝑏=1) as well as in the case of noncommutative algebras (𝑏≠1), and that a noncommutative algebra with 𝑏=−1 exists, where the solutions take the most simple structure.

Consider, furthermore, that Theorem 3.3 is not a simple generalization of Theorem 3.1 by generalizing the dimension and introducing a nonvanishing stress energy tensor. Theorem 3.1 works in the framework of 𝔗¹(𝑊;∗)_(𝜉0), whereas Theorem 3.3 works in the framework of 𝔗₁(𝑊;∗)_(𝜉0).

It remains to show that the metric in Theorem 3.3 differs from the metric presented in [17]. For this we compare the following: Metricin[17]∶𝑔_𝑖𝑗(𝜉)=𝜂_𝑖𝑗+(𝐶^𝑙_{(∗)𝑖𝑗}+𝐶^𝑙_{(∗)𝑗𝑖})𝐴_𝑙(𝜉),witharbitrarysplittingparametersMetricabove∶𝑔_𝑖𝑗(𝜉)=(1+⟨𝑤,𝔇_(∗)𝑓⟩)𝜂_𝑖𝑗+(𝐶^𝑙_{(∗)𝑖𝑗}+𝐶^𝑙_{(∗)𝑗𝑖})𝑎_𝑙(𝜉)withrestrictedsplittingparameters𝑑_𝑖𝑑_𝑗𝜂^𝑖𝑗=0, (3.17) with 𝐴,𝑎,𝑓∈𝔗₁(𝑊;∗)_(𝜉0). The metrics become identical if we restrict the algebraic framework of the first metric by 𝑑_𝑖𝑑_𝑗𝜂^𝑖𝑗=0 and the second metric by 𝔇_(∗)𝑓=0; in all other cases the metrics are different.

Finally some remarks to the physical importance of the stress energy tensor above are presented. For this we have a look on the following overview:𝑇stressenergytensorphysicalcharacterization_𝑖𝑗=(𝑝+𝜀)𝑢_𝑖𝑢_𝑗+𝑝𝑔_𝑖𝑗𝑇macroskopicmatter_𝑖𝑗=14𝜋(𝐹_𝑖𝑚𝐹^𝑚_𝑗−14𝐹_𝑝𝑞𝐹^𝑝𝑞𝑔_𝑖𝑗)with𝐹_𝑖𝑚∶=𝐴_𝑚;𝑖−𝐴_𝑖;𝑚𝑇freeelectromagneticstressenergytensor_𝑖𝑗=𝜇₀𝑢_𝑖𝑢_𝑗+14𝜋(𝐹_𝑖𝑚𝐹^𝑚_𝑗−14𝐹_𝑝𝑞𝐹^𝑝𝑞𝑔_𝑖𝑗)incoherentmatterintheelectromagneticfield. (3.18a)(3.18b)(3.18c)

(i)The case (3.18a) describes macroscopic matter with velocity 𝑢_𝑖, pressure 𝑝, and energy density 𝜀.(ii)The case (3.18b) describes the interchanging of the space-time structure with the electromagnetic field, where 𝐹 denotes the antisymmetric electromagnetic field tensor, defined by the potential 𝐴.(iii)The case in (3.18c) describes the interchanging of the space-time structure with incoherent matter in the electromagnetic field.

The stress energy tensor considered in Theorem 3.3 is defined by 𝑇_𝑖𝑗=𝜀𝑑_𝑖𝑑_𝑗, which allows a physical interpretation as macroscopic matter with velocity 𝑑_𝑖 (constancy by introducing a special coordinate system), energy density 𝜀, and pressure 𝑝=0. It can be shown that also special electromagnetic fields lead to the stress energy tensor above.

3.2. New Explicit Solutions of Navier-Stokes Equations

“Although these [Navier-Stokes] equations were written down in the 19th century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.”

With these words, the Clay Mathematic Institute describes a problem which is denoted as one of the seven “Millennium Problems” (see at http://www.claymath.org/). In the official formulation (The complete formulation can be found for instance in http://www.claymath.org./) by [20], the Navier-Stokes equations are presented as follows.

“The Euler and Navier-Stokes equations describe the motion of a fluid in ℝ^𝑛(𝑛=2 or 3). These equations are to be solved for an unknown velocity vector 𝑢(𝑥,𝑡)=(𝑢_𝑖(𝑥,𝑡))_{1≤𝑖≤𝑛}∈ℝ^𝑛 and pressure 𝑝(𝑥,𝑡)∈ℝ, defined for position 𝑥∈ℝ^𝑛 and time 𝑡≥0. We restrict attention here to incompressible fluids filling all of ℝ^𝑛. The Navier-Stokes equations are then given by

𝜕𝑢𝜕𝑡^𝑖+^𝑛_𝑗=1𝑢_𝑗𝜕𝑢_𝑖𝜕𝑥_𝑗=𝜈Δ𝑢_𝑖−𝜕𝑝𝜕𝑥_𝑖+𝑓_𝑖(𝑥,𝑡),𝑥∈ℝ^𝑛,𝑡≥0,div𝑢=^𝑛_𝑗=1𝜕𝑢_𝑖𝜕𝑥_𝑖=0,𝑥∈ℝ^𝑛,𝑡≥0, (3.19a)(3.19b) with initial conditions 𝑢(𝑥,0)=𝑢⁰(𝑥),𝑥∈ℝ^𝑛. (3.20)

Here, 𝑢⁰(𝑥) is a given, 𝐶^∞ divergence-free vector field on ℝ^𝑛,𝑓_𝑖(𝑥,𝑡) are the components of a given, externally applied force (e.g., gravity), 𝜈 is a positive coefficient (the viscosity), and Δ=∑^𝑛_𝑗=1(𝜕²/𝜕𝑥²_𝑗)  is the Laplacian in the space variables…”

Before presenting our new GAP-solution method, we remember some standard methods and main results of Navier-Stokes solution theory. Leray [21] has introduced the program of analyzing weak solutions of Navier-Stokes equations and he was able to show that in the three-dimensional case always exists a weak solution with suitable growth properties. Lerays method was taken and amended by Scheffer [22], Caffarelli et al. [23], Lin [24], Shnirelman [25], and others with the main result that the growth conditions of weak solutions can be classified quantitative [23, 24].

But at time it is not possible to perform the step from weak solutions to the physically interesting smooth solutions; we do not know if smooth initial conditions always induce smooth solutions or not. Also uniqueness of weak solutions is not known. Furthermore, the concrete construction of Navier-Stokes-solutions is a very hard problem. Standard methods for a nonnumerical construction are group methods, also the classical method of separation is used in a modified way, but actually the gained results are very small, which is not a surprise; all  nonnumerical methods deal with very symmetric structures, and we cannot expect that such special tools delivers very general results.

In the official formulation [20] of the “Navier-Stokes Millenium Problem”, these difficulties are summarized as follows:

“⋯our understand [of the Navier-Stokes equations] is a very primitive level. Standard methods from PDE appear inadequate to settle the problem.”

In the following, we will rewrite the Navier-Stokes system (3.19a) and (3.19b). With (In the following considerations, always a cartesian coordinate system is assumed. Thus the contravariant components 𝑢^𝑖 of the vector field 𝑢 can be identified with the covariant components 𝑢_𝑖 from [20] in (3.19a) and (3.19b)). 𝑢^𝑖_,𝑗∶=𝜕𝑢^𝑖/𝜕𝑥^𝑗,  𝑞^𝑖∶=−𝑝^,𝑖+𝑓^𝑖, the Einstein summation convention, and the generalization of ℝ^𝑛 to a region 𝑊⊆ℝ^𝑛 (in the case 𝑊≠ℝ^𝑛, adequate boundary conditions must be assumed), we get

𝑢^𝑖_,𝑡−𝜈Δ𝑢^𝑖+𝑢^𝑗𝑢^𝑖_,𝑗=𝑞^𝑖,𝑢^𝑖_,𝑖=0,for𝑥∈𝑊,𝑡≥0. (3.21a)(3.21b) We will solve these equations under the assumptions 𝑢(𝑥,𝑡)=𝑈(𝑧(𝑥,𝑡)),  𝑞(𝑥,𝑡)=𝑄(𝑧(𝑥,𝑡)), where the 𝑛-dimensional vector fields 𝑈(𝑧) and 𝑄(𝑧) are assumed as ∗-analytic in the 𝑛-dimensional vector 𝑧 of the following structure: 𝑧(𝑥,𝑡)=𝜆∗𝑥+𝜇𝑡,𝑊⊆ℝ^𝑛,𝑡∈[0,∞[,𝑧∈𝑊_𝑧. (3.22) Here 𝜆,𝜇∈ℝ^𝑛 denote arbitrary fixed constants and 𝑊_𝑧⊆ℝ^𝑛 denotes the region of 𝑧.

Theorem 3.5. Given a commutative pseudoring (ℝ^𝑛;∗)with structure constants 𝐶^𝑘_{(∗)𝑖𝑗},𝐶^𝑚_(∗)[3]∶=𝐶^𝑖𝑚_(∗)𝑖,  𝐶_(∗)[1]𝑚∶=𝐶^𝑖_{(∗)𝑚𝑖}, and further given two arbitrary vectors 𝜆,𝜇∈ℝ^𝑛, the 𝑛-dimensional Navier-Stokes system (3.21a) and (3.21b) with viscosity coefficient𝜈∈ℝand an arbitrary inhomogeneity of the symmetry 𝑞(𝑥,𝑡)=𝑄(𝑧(𝑥,𝑡)),𝑄∈𝔗¹(𝑊_𝑧;∗)_(𝑧0)anarbitraryfield (3.23) with 𝑧(𝑥,𝑡) from (3.22), then the following statements hold.Statement 1. A solution 𝑢(𝑥,𝑡) of the Navier-Stokes differential equations (3.21a) is given by (a)𝑢(𝑥,𝑡)=𝑈(𝑧(𝑥,𝑡)),(b)𝑈(𝑧)=−2𝜈𝜆∗𝐶_(∗)[3]∗𝑌^−1(∗)(𝑧)∗𝔇_(∗)𝜈𝑌,𝑌(𝑧)definedbythelineardifferentialequation,(c)²(𝐶_(∗)[3])²∗𝜆³∗𝔇²_(∗)𝑌−𝜈𝐶_(∗)[3]∗𝜆∗𝜇∗𝔇_(∗)1𝑌−2(𝔇⁻¹_(∗)𝑄)∗𝑌=0. (3.24)Statement 2. If the inhomogeneity 𝑄 satisfies the condition 𝐶^[𝑗𝑘]_(∗)𝑖(𝔇_(∗)𝑄)^𝑖=0, (3.25) it becomes interpretable as the pressure gradient 𝑄=grad𝑝, and therefore the pressure is explicitly given by 𝑝(𝑥,𝑡)=𝑄_𝑖(𝑧(𝑥,𝑡))𝑑𝑥^𝑖. (3.26)Statement 3. If furthermore 𝐶_(∗)[1]𝑖𝐶^𝑘_{(∗)𝑖𝑗}𝜆^𝑗=0, (3.27) also the divergence condition (3.21b) is satisfied.

Before considering the proof structure (the extensive proof is omitted for space reason), we will give some hints to the statements above from a physical point of view as well as from a mathematical point of view. We begin by remembering that the entire Navier-Stokes-problem as formulated in (3.19a) and (3.19b) can be split up to three subproblems:

(1)the problem of solving the Navier-Stokes differential equation (3.19a) for an arbitrary given inhomogeneity (−𝜕𝑝/𝜕𝑥_𝑖+𝑓_𝑖 must not be a gradient),(2)the problem of restricting (3.19a) solutions by demanding inhomogeneities of gradient form.(3)the problem of restricting (3.19a)-solutions (for gradient inhomogenities as well as for arbitrary inhomogeneities) by demanding the divergence condition (3.19b).

The solution of problem 1 is interesting from a mathematical point of view and the solution of problem 2 is interesting not only from a mathematical but also from a physical point of view, because for many practical applications, gradient inhomogeneities (pressure gradient) are assumed. Finally the solution of problem 3 (for gradient inhomogeneities) is interesting for getting a prize (see the “Millennium Problem”).

Our statements in the Theorem 3.5 follow this classification: Statement 1 solves the problem 1 (without any restriction on the used algebraic framework), Statement 2 solves the problem 2 (by restricting the algebraic framework according to (3.25)), and Statement 3 gives a condition for solving the remaining problem 3 (by a further restriction of the algebraic framework according to (3.27)). It can be shown explicitly that algebras exist, such that the problems 1 and 2 can be solved explicitly (Kronecker-algebras, see the Section 2), but at time we do not know an algebra such that all three problems become solvable at the same time (for a new, unknown solution set).

Therefore, the statements of Theorem 3.5 are interesting from a mathematical point of view (Statement 1) as well as from a physical point of view (Statement 2), but—of course—the Millennium Problem remains unsolved.

Now some remarks to the solution structure (3.21a) and (3.21b) are presented.

(i)The solution structure holds for arbitrary commutative pseudorings; we are not restricted on splitting pseudorings as in the case of the Einstein solutions above.(ii)The vector 𝑌^−1(∗)(𝑧) appears, that is, the solution 𝑈(𝑧)generally will have singularities. In division algebras, these singularities are described entirely by values 𝑧_𝛼, 𝛼=1,2,… with the property 𝑌^−1(∗)(𝑧_𝛼)=0, but in nondivision algebras there exist a larger set of singularities (see Section 2 and also the following considerations). In other words, the solution structure above allows an explicit description of the singularities depending on the chosen algebra.(iii)It must be mentioned that our considerations here were restricted on the simple cases of commutative pseudorings and the special 𝔗¹-symmetry. Noncommutative pseudorings and generalizations of 𝔗¹-symmetry (or 𝔗₁-symmetry) lead to very complex Navier-Stokes solutions which cannot be presented in this framework.

Finally some hints to the proof-structure are presented.

(1)It is shown that in the case of a commutative pseudoring (which is assumed) the assumed symmetry 𝑄(𝑧)∈𝔗¹(𝑊_𝑧;∗)_(𝑧0) of the inhomogeneity leads to the symmetry 𝑢(𝑥,𝑡)=𝑈(𝑧)∈𝔗¹(𝑊_𝑧;∗)_(𝑧0).(2)This property of 𝑈 now allows the transformation of the Navier-Stokes differential equation (3.21a) onto a ”GAP-Riccati differential equation” (i.e., an algebra valued Riccati equation).(3)The assumed commutativity of the pseudoring now allows to solve this GAP-Riccati equation in the same way as the classical scalar Riccati-equation by transforming this equation onto a linear (algebra-valued) second-order differential equation (see (3.24)(c)).(4)The explicit solution of this linear differential equation by comparison of coefficients is a simple problem, which only depends on the concrete structure of 𝔇⁻¹_(∗)𝑄, that is, on the structure of the given inhomogeneity 𝑄.(5)After calculating the vector field 𝑌(𝑧) from (3.24)(c), we get the vector field 𝑈(𝑧) according to (3.24)(b). The inverse vector field 𝑌^−1(∗)(𝑧) is calculable explicitly for the physical interesting dimensions 𝑛≤3, in the case of a splitting pseudoring (see Section 2) also for arbitrary dimensions 𝑛, and therefore, 𝑈(𝑧) is calculable explicitly.(6)After calculating the vector field 𝑈(𝑧), we get the Navier-Stokes solution 𝑢(𝑥,𝑡) according to (3.24)(a) by inserting 𝑧(𝑥,𝑡)=𝜆∗𝑥+𝜇𝑡 (see (3.22)).

3.3. The Necessity of an Embedding Formalism

The results above show that GAP-Theory allows the explicit solution of important nonlinear partial differential equations from Theoretical Physics. For space reason, we have restricted these applications on Einstein field equations and Navier-Stokes equations, but GAP-Theory also allows applications on other famous differential equations as the Yang-Mills equation, the Korteweg-de Vries equation, and so forth.

The fundamental symmetry-framework of our considerations was the symmetry of ∗-analyticity (GAP-symmetry) which has turned out as “compatible” with the considered partial differential equations in the sense that it allows an explicit solution formalism. From the mathematical point of view, ∗-analyticity seems to be a useful tool to look inside the structure of covariant differential equations, but from the physical point of view ∗-analyticity represents a strong restriction: in many cases problem defining physically vector fields (tensor fields) will be not be ∗-analytical but only smooth.

To demonstrate the concrete problems, we have a look on the Theorem 3.5 for solving the Navier-Stokes system (3.21a) and (3.21b) in the framework of ∗-analyticity: Theorem 3.5 shows that a ∗-analytical inhomogeneity 𝑞 allows an explicit solution of the Navier-Stokes differential equation (3.21a) for arbitrary commutative pseudorings, and that furthermore for a special set of these pseudorings also the divergence condition (3.21b) holds. But for many applications, the inhomogeneity 𝑞 will not be ∗-analytical but a gradient, and in this case Theorem 3.5 is applicable only on ∗-analytical gradients, which represents a strong restriction.

Similar considerations hold for the Einstein equations: In Theorem 3.3  ∗-analyticity appears in the stress energy tensor (in a hidden form) as well as in the metric tensor, but a lot of physical interesting stress energy tensors will not be ∗-analytical.

These considerations boil down to the following question: is it possible to embed an arbitrary 𝑛-dimensional smooth vector field 𝑓 (tensor field) into an 𝑁-dimensional ∗-analytical vector field 𝐹 (tensor field) for a sufficient large dimension 𝑁≥𝑛 in the sense, that the first 𝑛 components of 𝐹 allow a representation of the 𝑛 components of 𝑓? (An exact definition of “embedding” will be given in Section 4.)

In the case of a positive answer, we can embed a 𝑛-dimensional nonlinear partial differential equation defined by only smooth vector fields onto a 𝑁-dimensional nonlinear partial differential equation defined by ∗-analytical vector fields, and this high-dimensional differential equation can be solved in the framework of GAP-Theory as demonstrated in the sections above.

Finally the first 𝑛 components of the obtained 𝑁-dimensional solution vector field (tensor field) represent the solution of the 𝑛-dimensional start problem.

Section 4 will show that such an Embedding formalism of smoothness into ∗-analyticity (or more general: chain analyticity) exists.

4. Embedding Problems for GAPs. New Products

Embedding problems (also denoted as Immersion problems) are important in mathematical physics as well as in nonlinear system theory [14, 16, 26, 27] and can be formulated in various ways. Roughly spoken immersion theory asks if an 𝑛-dimensional structure with small symmetry can be seen as a “projection” of an 𝑁-dimensional structure with high symmetry, 𝑛≤𝑁. Of course the answer depends on the concretely given symmetry classes as well as on the dimensions 𝑛 and 𝑁.

In the following, we are interested on a special class of embedding problems, defined by the question if smoothness (which is a very general symmetry) can be embedded into chain-analyticity (which is a rather special symmetry).

Definition 4.1. Given an 𝑛-dimensional smooth vector field 𝑓∈𝔗¹(𝑊^𝑛), 𝑊^𝑛⊆ℝ^𝑛, and an 𝑁-dimensional chain-analytic vector field 𝐹∈𝔗¹(𝑊^𝑁;∗₁,…,∗_𝑀)_(𝑧0), 𝔛=(ℝ^𝑁;∗₁,…,∗_𝑀) an arbitrary PA1-chain, then 𝑓 is denoted as embaddable into𝐹 if it holds that 𝑓^𝛾(𝑧)=𝐹^𝛾(𝑍)∣_𝑍^𝑛+1_=𝑍^𝑛+2_⋯=𝑍^𝑁₌₀,with𝑍^𝛾=𝑧^𝛾,𝛾∈𝐼_1,𝑛. (4.1) If this relation holds for arbitrary 𝑓∈𝔗¹(𝑊^𝑛), then 𝔗¹(𝑊^𝑛) is denoted as embaddable into 𝔗¹(𝑊^𝑁;∗₁,…,∗_𝑀)_(𝑧0).

Of course the possibility of embedding 𝔗¹(𝑊^𝑛) into 𝔗¹(𝑊^𝑁;∗₁,…,∗_𝑀)_(𝑧0) will depend on the concrete form of the given PA1-chain 𝔛 as well as on the proportion of 𝑛 to 𝑁. Is 𝔗¹(𝑊^𝑛) embaddable for 𝑁-dimensional Kronecker PA1-chains, 𝑁 finite and/or for 𝑁-dimensional Splitting PA1-chains, 𝑁 finite? Or must 𝑁 be infinite? We will answer these questions by specializing on the subset 𝔗¹_[𝑃](𝑊^𝑛)⊆𝔗¹(𝑊^𝑛) defined by all polynomials of order 𝑝≤𝑃∈ℕ, and by specializing PA1-chains to pseudorings.

Theorem 4.2. Given the following classes of 𝑁-dimensional pseudorings 𝔗¹(ℝ^𝑁;∗)_(𝑧0): 𝐶^𝑘_{(∗)𝑖𝑗}(𝜇)=𝛿^𝑘_{𝐻𝑁(𝑖+𝐻𝑁(𝑢𝜇(𝑗)))},𝑢_𝜇(𝑗)∶=𝑗^𝜇,𝜇>²log(𝑛+1),𝜇∈ℕ,𝑖,𝑗,𝑘∈𝐼_1,𝑁, (4.2) with the index function 𝐻_𝑁 from (2.55), then the following statements hold. Statement 1. For every given pair 𝑛,𝑃∈ℕthere exist finite numbers 𝑁(𝑛,𝑃,𝜇)∈ℕsuch that 𝔗¹_[𝑃](𝑊^𝑛)is embaddable into 𝔗¹_[𝑃](𝑊^𝑁;∗)₍₀₎,0∈𝑊^𝑛⊂𝑊^𝑁.Statement 2. A sufficient condition for embedding is given by 𝑁≥𝑁_min(𝑛,𝑃,𝜇),with𝑁_min(𝑛,𝑃,𝜇)∶=𝑛+𝑛^𝜇𝑃. (4.3)Statement 3. Let 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝} with 𝛾,𝛽₁,…,𝛽_𝑝∈𝐼_1,𝑛,𝑝∈𝐼_0,𝑃 denote the coefficients of an arbitrary element 𝑓∈𝔗¹_[𝑃](𝑊^𝑛).Then under the assumption (4.3) the coefficients 𝐴^𝑖_[𝑝],𝑝∈𝐼_0,𝑃, 𝑖∈𝐼_1,𝑁of the corresponding element 𝐹∈𝔗¹_[𝑃](𝑊^𝑁;∗)₍₀₎are defined by 𝐴^𝛾_[0]=𝑎^𝛾_[0],𝐴^{𝛾+∑𝑝𝑘=1(𝛽𝑘)𝜇}_[𝑝]=𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝},𝑝∈𝐼_1,𝑃. (4.4)

Proof
Step 1. The pseudoring symmetry of 𝔗¹(ℝ^𝑁;∗)_(𝑧0)∶ for a given 𝜇, it is 𝐶^𝑙_{(∗)𝑖𝑗}(𝜇)𝐶^𝑚_{(∗)𝑙𝑘}(𝜇)=𝛿^𝑙_{𝐻𝑁(𝑖+𝐻𝑁(𝑢(𝑗)))}𝛿^𝑚_{𝐻𝑁(𝑙+𝐻𝑁(𝑢(𝑘)))}=𝛿^𝑚_{𝐻𝑁(𝐻𝑁(𝑖+𝐻𝑁(𝑢(𝑗)))+𝐻𝑁(𝑢(𝑘)))}=𝛿^𝑚_{𝐻𝑁(𝑖+𝐻𝑁(𝑢(𝑗))+𝑢(𝑘))}=𝛿^𝑚_{𝐻𝑁(𝑖+𝑢(𝑗)+𝑢(𝑘))},⇒𝐶^𝑙_{(∗)𝑖[𝑗}(𝜇)𝐶^𝑚_{(∗)|𝑙|𝑘]}(𝜇)=0, (4.5) where we have used the property 𝐻_𝑁(𝐻_𝑁(𝑘)+𝑚)=𝐻_𝑁(𝑘+𝑚), and so forth.Step 2. The conditions of embaddability 𝔗¹_[𝑃](𝑊^𝑛)into 𝔗¹_[𝑃](𝑊^𝑁;∗₁,…,∗_𝑃)₍₀₎in general PA1-chains: By definition every element 𝑓∈𝔗¹_[𝑃](𝑊^𝑛),0∈𝑊^𝑛 has a representation (In the following we use Greek indices for the range 𝐼_1,𝑛, Latin indices for the ranges 𝐼_1,𝑁 and 𝐼_0,𝑃.) 𝑓^𝛾(𝑧)=𝑎^𝛾_[0]+𝑎^𝛾_[1]𝛽1𝑧^𝛽1+𝑎^𝛾_{[2]𝛽1𝛽2}𝑧^𝛽1𝑧^𝛽2+⋯+𝑎^𝛾_{[𝑃]𝛽1𝛽2⋯𝛽𝑃}𝑧^𝛽1𝑧^𝛽2⋯𝑧^𝛽𝑃, (4.6)𝑧∈𝑊^𝑛, where the coefficients 𝑎^𝛾_{[𝑝]𝛽1⋯𝛽𝑝}∈ℝ for  𝛾,𝛽₁,…,𝛽_𝑝∈𝐼_1,𝑛,  𝑝∈𝐼_0,𝑃 can be assumed as total symmetric in the indices 𝛽₁,…,𝛽_𝑝. Furthermore, for a general given PA1-chain 𝔛=(ℝ^𝑁;∗₁,…,∗_𝑃) every element 𝐹∈𝔗¹_[𝑃](𝑊^𝑁;∗₁,…,∗_𝑃)₍₀₎ has a representation 𝐹(𝑍)=𝐴_[0]+𝐴_[1]∗₁𝑍+(𝐴_[2]∗₂𝑍)∗₁𝑍+((𝐴_[3]∗₃𝑍)∗₂𝑍)∗₁𝑍+⋯+(⋯((𝐴_[𝑀]∗_𝑃𝑍)∗_𝑃−1𝑍)∗_𝑃−2⋯)∗₁𝑍,𝑍∈𝑊^𝑁 (4.7) with certain vector-valued coefficients 𝐴_[𝑝]∈ℝ^𝑁,  𝑝∈𝐼_0,𝑃. In component representation, 𝐹^𝑘(𝑋)=𝐴^𝑘_[0]+𝐴^𝑖_[1]𝐶^𝑘_{(∗1)𝑖𝑗}𝑍^𝑗+𝐴^𝑖_[2]𝐶^𝑙1_{(∗2)𝑖𝑗1}𝑍^𝑗1𝐶^𝑘_{(∗1)𝑙1𝑗2}𝑍^𝑗2+⋯+𝐴^𝑖_[𝑃]𝐶^𝑙1_{(∗𝑃)𝑖𝑗1}𝑍^𝑗1𝐶^𝑙2_{(∗𝑃−1)𝑙1𝑗2}𝑍^𝑗2⋯𝐶^𝑘_{(∗1)𝑙𝑃−1𝑗𝑃}𝑍^𝑗𝑃_rewritten=𝐴^𝑘_[0]+𝐴^𝑖_[1]𝐶^𝑘_{(∗1)𝑖𝑗}𝑍^𝑗+𝐴^𝑖_[2]𝐶^𝑙1_{(∗2)𝑖𝑗1}𝐶^𝑘_{(∗1)𝑙1𝑗2}𝑍^𝑗1𝑍^𝑗2+⋯+𝐴^𝑖_[𝑃]𝐶^𝑙1_{(∗𝑃)𝑖𝑗1}𝐶^𝑙2_{(∗𝑃−1)𝑙1𝑗2}⋯𝐶^𝑘_{(∗1)𝑙𝑃−1𝑗𝑃}𝑍^𝑗1𝑍^{𝑗2𝑗𝑃}. (4.8) The restriction on the subset 𝑊^𝑛⊆𝑊^𝑁 by 𝑍^𝑛+1=𝑍^𝑛+2⋯=𝑍^𝑁=0,𝑍^𝛼=𝑧^𝛼,𝛼∈𝐼_1,𝑛 leads to 𝐹^𝛾(𝑍)|_𝑍^𝑛+1_=𝑍^𝑛+2_⋯=𝑍^𝑁₌₀=𝐴^𝛾_[0]+𝐴^𝑖_[1]𝐶^𝛾_{(∗1)𝑖𝛽1}𝑧^𝛽1+𝐴^𝑖_[2]𝐶^𝑙1_{(∗2)𝑖𝛽1}𝐶^𝛾_{(∗1)𝑙1𝛽2}𝑧^𝛽1𝑧^𝛽2+⋯+𝐴^𝑖_[𝑃]𝐶^𝑙1_{(∗𝑃)𝑖𝛽1}𝐶^𝑙2_{(∗𝑃−1)𝑙1𝛽2}⋯C^𝛾_{(∗1)𝑙𝑃−1𝛽𝑃}𝑧^𝛽1𝑧^𝛽2⋯z^𝛽𝑃,𝛾∈𝐼_1,𝑛, (4.9) and the embedding condition (4.1) takes the form 𝑎^𝛾_[0]+𝑎^𝛾_[1]𝛽1𝑧^𝛽1+𝑎^𝛾_{[2]𝛽1𝛽2}𝑧^𝛽1𝑧^𝛽2+⋯+𝑎^𝛾_{[𝑃]𝛽1𝛽2⋯𝛽𝑃}𝑧^𝛽1𝑧^𝛽2⋯𝑧^𝛽𝑃=𝐴^𝛾_[0]+𝐴^𝑖_[1]𝐶^𝛾_{(∗1)𝑖𝛽1}𝑧^𝛽1+𝐴^𝑖_[2]𝐶^𝑙1_{(∗2)𝑖𝛽1}𝐶^𝛾_{(∗1)𝑙1𝛽2}𝑧^𝛽1𝑧^𝛽2+⋯+𝐴^𝑖_[𝑃]𝐶^𝑙1_{(∗𝑃)𝑖𝛽1}𝐶^𝑙2_{(∗𝑃−1)𝑙1𝛽2}𝐶^𝛾_{(∗1)𝑙𝑃−1𝛽𝑃}𝑧^𝛽1𝑧^𝛽2⋯𝑧^𝛽𝑃,𝛾∈𝐼_1,𝑛. (4.10) Comparison of the components leads to the following equation system: 𝐴^𝛾_[0]=𝑎^𝛾_[0],𝐴^𝑖_[𝑝]𝐶^𝑙1_{(∗𝑝)𝑖𝛽1}𝐶^𝑙2_{(∗𝑝−1)𝑙1𝛽2}⋯𝐶^𝛾_{(∗1)𝑙𝑝−1𝛽𝑝}=𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝},(b), (4.11)𝛾,𝛽₁,…,𝛽_𝑝∈𝐼_1,𝑛,  𝑝∈𝐼_1,𝑃,  or in detailed denotion: 𝐴^𝛾_[0]=𝑎^𝛾_[0],𝐴^𝑖_[1]𝐶^𝛾_{(∗1)𝑖𝛽1}=𝑎^𝛾_[1]𝛽1,𝐴^𝑖_[2]𝐶^𝑙1_{(∗2)𝑖𝛽1}𝐶^𝛾_{(∗1)𝑙1𝛽2}=𝑎^𝛾_{[2]𝛽1𝛽2},𝐴^𝑖_[3]𝐶^𝑙1_{(∗3)𝑖𝛽1}𝐶^𝑙2_{(∗2)𝑙1𝛽2}𝐶^𝛾_{(∗1)𝑙2𝛽3}=𝑎^𝛾_{[3]𝛽1𝛽2𝛽3},⋮𝐴^𝑖_[𝑃]𝐶^𝑙1_{(∗𝑃)𝑖𝛽1}𝐶^𝑙2_{(∗𝑃−1)𝑙1𝛽2}⋯𝐶^𝛾_{(∗1)𝑙𝑃−1𝛽𝑃}=𝑎^𝛾_{[𝑃]𝛽1𝛽2⋯𝛽𝑃}, (4.12)𝛾,𝛽₁,…,𝛽_𝑝∈𝐼_1,𝑛. We see that the total symmetry in the covariant indices of the right side is compatible with the total symmetry in the covariant indices of the left side which follows immediately from the PA1-chain symmetry. The question is now if this system is solvable for arbitrary given 𝑛,𝑃 and components 𝑎^𝛾_{[𝑃]𝛽1⋯𝛽𝑝},𝛾,𝛽₁,…,𝛽_𝑝∈𝐼_1,𝑛,𝑝∈𝐼_0,𝑃, that is, if we can find an adequate dimension 𝑁(𝑛,𝑃), adequate coefficients 𝐴^𝑖_[𝛼],𝛼∈𝐼_0,𝑃,𝑖∈𝐼_1,𝑁, and an adequate PA1-chain with structure constants 𝐶^𝑘_{(∗𝛼)𝑖𝑗} such that (4.12) get solvable. Since not only the coefficients 𝐴^𝑖_[𝛼] but also the structure constants are unknown, the equation system is nonlinear.Step 3. The conditions of embaddability 𝔗¹_[𝑃](𝑊^𝑛)into 𝔗¹_[𝑃](𝑊^𝑁;∗)₍₀₎in the special PA1-chain (4.2): now we have to insert the pseudoring structure (4.2) in the equation system above. For this we calculate 𝐶^𝑙1_{(∗2)𝑖𝛽1}𝐶^𝛾_{(∗1)𝑙1𝛽2}=𝛿^𝑙1_{𝐻𝑁(𝑖+𝐻𝑁(𝑢𝜇(𝛽1)))}𝛿^𝛾_{𝐻𝑁(𝑙1+𝐻𝑁(𝑢𝜇(𝛽2)))}=𝛿^𝛾_{𝐻𝑁(𝐻𝑁(𝑖+𝐻𝑁(𝑢𝜇(𝛽1)))+𝐻𝑁(𝑢𝜇(𝛽2)))seestep1above}=𝛿^𝛾_{𝐻𝑁(𝑖+𝑢𝜇(𝛽1)+𝑢𝜇(𝛽2))},𝐶^𝑙1_{(∗3)𝑖𝛽1}𝐶^𝑙2_{(∗2)𝑙1𝛽2}𝐶^𝛾_{(∗1)𝑙2𝛽3}=𝛿^𝑙1_{𝐻𝑁(𝑖+𝐻𝑁(𝑢𝜇(𝛽1)))}𝛿^𝛾_{𝐻𝑁(𝑙1+𝑢𝜇(𝛽2)+𝑢𝜇(𝛽3))}=𝛿^𝛾_{𝐻𝑁(𝐻𝑁(𝑖+𝐻𝑁(𝑢𝜇(𝛽1)))+𝑢𝜇(𝛽2)+𝑢𝜇(𝛽3))seestep1above}=𝛿^𝛾_{𝐻𝑁(𝑖+𝑢𝜇(𝛽1)+𝑢𝜇(𝛽2)+𝑢𝜇(𝛽3))}, (4.13) and we obtain for the general case 𝐶^𝑙1_{(∗𝑝)𝑖𝛽1}𝐶^𝑙2_{(∗𝑝−1)𝑙1𝛽2}⋯𝐶^𝛾_{(∗1)𝑙𝑝−1𝛽𝑝}=𝛿^𝛾_{𝐻𝑁(𝑖+𝑢𝜇(𝛽1)+𝑢𝜇(𝛽2)+⋯+𝑢𝜇(𝛽𝑝))inserting𝑢𝜇(𝛽𝑚)}=𝛿^𝛾_{𝐻𝑁(𝑖−(𝛽1)𝜇−(𝛽2)𝜇−⋯−(𝛽𝑝)𝜇)}=𝛿^𝛾_{𝐻𝑁(𝑖−∑𝑝𝑘=1(𝛽𝑘)𝜇)},𝑝∈𝐼_1,𝑃. (4.14) Inserting in the equation system (4.11), we get𝐴^𝛾_[0]=𝑎^𝛾_[0],𝐴^{𝐻𝑁(𝛾+∑𝑝𝑘=1(𝛽𝑘)𝜇)}_[𝑝]=𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝}, (4.15a)(4.15b)𝛾,𝛽₁,𝛽₂,…,𝛽_𝑝∈𝐼_1,𝑛,  𝑝∈𝐼_1,𝑃. Now we have to show that for arbitrary given 𝑛,𝑃 and arbitrary components 𝑎^𝛾_{[𝑃]𝛽1⋯𝛽𝑝}, there exist natural numbers 𝜇(𝑛,𝑃) and 𝑁(𝑛,𝑃,𝜇) and coefficients 𝐴^𝑖_[𝛼],  𝛼∈𝐼_0,𝑃,  𝑖∈𝐼_1,𝑁 with the property above. The positive answer will be given in the next step.Step 4. The solvability of the equation system (4.15a) and (4.15b): for the following considerations, we define the set 𝐼^[𝑝+1]_1,𝑛∶={(𝛾,𝛽₁,𝛽₂,…,𝛽_𝑝)with𝛾,𝛽_𝑖∈𝐼_1,𝑛,𝛽_𝑖+1≥𝛽_𝑖},𝑝∈𝐼_1,𝑃, (4.16) that is, 𝐼^[𝑝+1]_1,𝑛 is a subset of the set 𝐼^[𝑝+1]_1,𝑛∶=𝐼_1,𝑛×𝐼_1,𝑛×⋯×𝐼_1,𝑛, (𝑝+1 times) defined by the additional property 𝛽_𝑖+1≥𝛽_𝑖. We see that 𝐼^[𝑝+1]_1,𝑛 is the definition region of the index field (𝛾,𝛽₁,𝛽₂,…,𝛽_𝑝) from the coefficients 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝} in (4.15b), since for these coefficients the total symmetry in the covariant indices hold by assumption. For the following, it is advantageous to write 𝛽 instead of (𝛽₁,𝛽₂,…,𝛽_𝑝) and (𝛾,𝛽) instead of (𝛾,𝛽₁,𝛽₂,…,𝛽_𝑝). With these denotions, we can formulate a necessary and sufficient condition for the solvability of (4.15a) and (4.15b) for arbitrary given 𝑛,𝑃, and for arbitrary given coefficients 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝} as follows: 𝐻_𝑁(𝛾+^𝑝_𝑘=1(𝛽_𝑘)^𝜇)≠𝐻_𝑁(𝛾+^𝑝_𝑘=1(𝛽_𝑘)^𝜇)for(𝛾,𝛽)≠(𝛾,𝛽), (4.17) with (𝛾,𝛽),(𝛾,𝛽)∈𝐼^[𝑝+1]_1,𝑛,𝑝∈𝐼_1,𝑃 (if the following relation is not satisfied, that is, if there exists for a fixed 𝑝 at least one pair of (𝑝+1)-tuples (𝛾,𝛽)≠(𝛾,𝛽) such that 𝐻_𝑁(𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇)=𝐻_𝑁(𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇), the condition (4.15b) would imply for the fixed 𝑝: 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝}=𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝},  which means a specialization by the coefficients 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝}). We have to show that this relation holds under our assumptions of 𝜇(𝑛,𝑃) and 𝑁(𝑛,𝑃,𝜇)≥𝑁_min(𝑛,𝑃,𝜇) as stated in (4.2), (4.3). In the case 𝑁(𝑛,𝑃,𝜇)≥𝑁_min(𝑛,𝑃,𝜇)=𝑛+𝑛^𝜇𝑃, it is 𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇≤𝑁_min(𝑛,𝑃,𝜇) for 𝑝∈𝐼_1,𝑃 which implies 𝐻_𝑁(𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇)=𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇 for 𝑝∈𝐼_1,𝑃. Inserting in (4.15a) and (4.15b), we obtain the statement (4.4). The solvability condition (4.17) takes the form 𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇≠𝛾+∑^𝑝_𝑘=1(𝛽_𝑘)^𝜇 or rewritten as ^𝑝_𝑘=1(𝛽_𝑘)^𝜇−^𝑝_𝑘=1(𝛽_𝑘)^𝜇≠𝛾−𝛾,for(𝛾,𝛽)≠(𝛾,𝛽), (4.18)(𝛾,𝛽),(𝛾,𝛽)∈𝐼^[𝑝+1]_1,𝑛,𝑝∈𝐼_1,𝑃. From the considerations above, we see that this condition is sufficient for the solvability of the the system (4.15b) for general given 𝑛,𝑃,𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝}. Since 𝛾−𝛾 moves in the region [−𝑛,𝑛], the condition is certainly satisfied by demanding |^𝑝_𝑘=1(𝛽_𝑘)^𝜇−^𝑝_𝑘=1(𝛽_𝑘)^𝜇|>𝑛,for𝛽≠𝛽, (4.19)𝛽,𝛽∈𝐼^[𝑝+1]_1,𝑛,𝑝∈𝐼_1,𝑃. The worst case of this new condition is given if only one component of 𝛽 differs from one component of 𝛽. Let 𝑘₀ denote the corresponding index, then we obtain from above the worst case in the form |(𝛽_𝑘0)^𝜇−(𝛽_𝑘0)^𝜇|>𝑛, for 𝛽_𝑘0≠𝛽_𝑘0. The worst case of this scenario is given by the minimal possible difference, that is, by 𝛽_𝑘0=1,𝛽_𝑘0=2. In this case 𝜇 is defined by the sufficient condition |1−2^𝜇|>𝑛 or 2^𝜇>𝑛+1 which is satisfied by 𝜇>²log(𝑛+1) as stated in (4.2).

I will denote this Theorem as the Pseudoring Embedding Theorem, and the pseudoring (4.2) as embedding pseudoring. The Theorem shows that a non ∗-analytic vector field always can be embedded into a higher-dimensional ∗-analytic vector field. Furthermore, this Theorem not only characterizes the possibility of embedding but also allows an explicit description of the practical embedding process. Some remarks for a deeper understanding are as follows.

(i)Since the map 𝑢_𝜇 in the embedding pseudoring is not a linear map, the embedding pseudoring is neither commutative nor associative: embedding is not compatible with commutativity and associativity.(ii)The embedding pseudoring is a Kronecker algebra, which leads to the question if also splitting pseudorings can be used for embedding. It can be shown that this is not possible in a general framework, that is, for embedding problems Kronecker algebras are more important than splitting algebras. Otherwise, the applications in [17] show that in the framework of solving partial differential equations the splitting algebras seem to be more important than Kronecker algebras. The upshot is that both algebra-types are important.(iii)The embedding condition (4.3) on the embedding dimension 𝑁≥𝑁_min is a sufficient condition. It is possible that there exist cases where a smaller dimension 𝑁≤𝑁_min is purposeful.(iv)Until now our embedding analyse was restricted on polynomials. It can be generalized on arbitrary smooth vector fields, but in this case the embedding dimension 𝑁 will not remain finite, according to the embedding relation (4.3). Thus it would be interesting to investigate if (4.3) is not only sufficient but also necessary.(v)Relation (4.4) shows that the coefficients 𝐴^𝑖_[𝑝] of the vector field 𝐹 are not defined entirely by the coefficients 𝑎^𝛾_{[𝑝]𝛽1𝛽2⋯𝛽𝑝} of the vector field 𝑓: a lot of coefficients 𝐴^𝑖_[𝑝] remain unrestricted. In other words, embedding from 𝑓 into 𝐹 does not define 𝐹 entirely but only partially.(vi)Until now our investigation was fixed on the case of vector-field embedding problems. More general is the case of tensor field embedding problems, which can be solved in an analogue way. An analyse is omitted here for space reason.

As a simple example, we consider the case of embedding an (𝑛=2)-dimensional polynomial of order 𝑃=3. Then the parameter 𝜇∈ℕ is restricted by the condition 𝜇>²log(𝑛+1), and the smallest natural solution is given by 𝜇=2. The embedding dimension 𝑁 is restricted by the condition 𝑁≥𝑁_min(𝑛,𝑃,𝜇)=𝑛+𝑛^𝜇𝑃=2+2²⋅3=14, and we take the smallest value 𝑁=14. According to (4.4), the coefficients 𝐴^𝑖_[𝑝] of the embedding polynomial 𝐹 take the form 𝑝=0∶𝛾=1∶𝐴¹_[0]=𝑎¹_[0],𝛾=2∶𝐴²_[0]=𝑎²_[0],𝑝=1∶𝛾=1∶𝐴²_[1]=𝑎¹_[1]1,𝐴⁵_[1]=𝑎¹_[1]2,𝛾=2∶𝐴³_[1]=𝑎²_[1]1,𝐴⁶_[1]=𝑎²_[1]2,𝑝=2∶𝛾=1∶𝐴³_[2]=𝑎¹_[2]11,𝐴⁶_[2]=𝑎¹_[2]12,𝐴⁹_[2]=𝑎¹_[2]22,𝛾=2∶𝐴⁴_[2]=𝑎²_[2]11,𝐴⁷_[2]=𝑎²_[2]12,𝐴¹⁰_[2]=𝑎²_[2]22,𝑝=3∶𝛾=1∶𝐴⁴_[3]=𝑎¹_[3]111,𝐴⁷_[3]=𝑎¹_[3]112,𝐴¹⁰_[3]=𝑎¹_[3]122,𝐴¹³_[3]=𝑎¹_[3]222,𝛾=2∶𝐴⁵_[3]=𝑎²_[3]111,𝐴⁸_[3]=𝑎²_[3]112,𝐴¹¹_[3]=𝑎²_[3]122,𝐴¹⁴_[3]=𝑎²_[3]222, (4.20) and according to the remarks above the remaining coefficients 𝐴^𝑖_[𝑝] can be chosen to zero, that is, 𝐴³_[0]=𝐴⁴_[0]=⋯=𝐴¹⁴_[0]=0,𝐴¹_[1]=𝐴⁴_[1]=𝐴⁷_[1]=𝐴⁸_[1]=⋯=𝐴¹⁴_[0]=0, and so forth.

Finally we will discuss a certain extension of our pseudoring-theory, which is interesting for certain embedding problems. We remember that until now we only have dealed with products ⊛_(∗)∶𝑉^𝑝×𝑉→𝑉^𝑝 and ⊛^dual_(∗)∶𝑉_𝑞×𝑉→𝑉_𝑞, which leads to the question if “similar” products 𝑉^𝑝×𝑉^𝑝→𝑉^𝑝 and 𝑉_𝑞×𝑉_𝑞→𝑉_𝑞 can be defined with similar properties. This will be done now.

Definition 4.3. Given an arbitrary algebra (𝑉;∗) with structure constant tensor 𝐶_(∗), then the∗-associated product of order 𝑝:⊠_(∗)∶𝑉^𝑝×𝑉^𝑝→𝑉^𝑝 is defined by (𝑋⊠_(∗)𝑌)^{𝑘1𝑘2⋯𝑘𝑝}∶=𝑋^{𝑖1𝑖2⋯𝑖𝑝}𝐶^𝑘1_{(∗)𝑖1𝑗1}𝐶^𝑘2_{(∗)𝑖2𝑗2}⋯𝐶^𝑘𝑝_{(∗)𝑖𝑝𝑗𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}, (4.21) and the dual-associated product of order 𝑞:⊠^dual_(∗)∶𝑉_𝑞×𝑉^𝑞→𝑉_𝑞 is defined by (𝑋⊠^dual_(∗)𝑌)_{𝑘1𝑘2⋯𝑘𝑞}∶=𝑋_{𝑖1𝑖2⋯𝑖𝑞}𝐶^𝑖1_{(∗)𝑘1𝑗1}𝐶^𝑖2_{(∗)𝑘2𝑗2}⋯𝐶^𝑖𝑞_{(∗)𝑘𝑞𝑗𝑞}𝑌^{𝑗1𝑗2⋯𝑗𝑞}. (4.22)

Proposition 4.4. Given a PA1-structure (𝑉;∗₁,∗₂), then the following symmetry holds: (𝑋⊠_(∗1)𝑌)⊠_(∗2)𝑍=(𝑋⊠_(∗1)𝑍)⊠_(∗2)𝑌,∀𝑋,𝑌,𝑍∈𝑉^𝑝. (4.23)

Proof. It is {(𝑋⊠_(∗1)𝑌)⊠_(∗2)𝑍}^{𝑚1𝑚2⋯𝑚𝑝}=(𝑋⊠_(∗1)𝑌)^{𝑘1𝑘2⋯𝑘𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑙1}𝐶^𝑚2_{(∗2)𝑘2𝑙2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑙𝑝}𝑍^{𝑙1𝑙2⋯𝑙𝑝}=𝑋^{𝑖1𝑖2⋯𝑖𝑝}𝐶^𝑘1_{(∗1)𝑖1𝑗1}𝐶^𝑘2_{(∗1)𝑖2𝑗2}⋯𝐶^𝑘𝑝_{(∗1)𝑖𝑝𝑗𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑙1}𝐶^𝑚2_{(∗2)𝑘2𝑙2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑙𝑝}𝑍^{𝑙1𝑙2⋯𝑙𝑝}_rewritten=𝑋^{𝑖1𝑖2⋯𝑖𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}𝑍^{𝑙1𝑙2⋯𝑙𝑝}𝐶^𝑘1_{(∗1)𝑖1𝑗1}𝐶^𝑘2_{(∗1)𝑖2𝑗2}⋯𝐶^𝑘𝑝_{(∗1)𝑖𝑝𝑗𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑙1}𝐶^𝑚2_{(∗2)𝑘2𝑙2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑙𝑝}. (4.24) Since 𝐶^𝑘𝛼_{(∗1)𝑖𝛼[𝑙𝛼}𝐶^𝑚𝛼_{(∗2)|𝑘𝛼|𝑗𝛼]}=0,𝛼∈𝐼_1,𝑝, the term above can be written as 𝑋^{𝑖1𝑖2⋯𝑖𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}𝑍^{𝑙1𝑙2⋯𝑙𝑝}𝐶^𝑘1_{(∗1)𝑖1𝑙1}𝐶^𝑘2_{(∗1)𝑖2𝑙2}⋯𝐶^𝑘𝑝_{(∗1)𝑖𝑝𝑙𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑗1}𝐶^𝑚2_{(∗2)𝑘2𝑗2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑗𝑝rewritten}=𝑋^{𝑖1𝑖2⋯𝑖𝑝}𝐶^𝑘1_{(∗1)𝑖1𝑙1}𝐶^𝑘2_{(∗1)𝑖2𝑙2}⋯𝐶^𝑘𝑝_{(∗1)𝑖𝑝𝑙𝑝}𝑍^{𝑙1𝑙2⋯𝑙𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑗1}𝐶^𝑚2_{(∗2)𝑘2𝑗2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑗𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}_rewritten=(𝑋⊠_(∗1)𝑍)^{𝑘1𝑘2⋯𝑘𝑝}𝐶^𝑚1_{(∗2)𝑘1𝑗1}𝐶^𝑚2_{(∗2)𝑘2𝑗2}⋯𝐶^𝑚𝑝_{(∗2)𝑘𝑝𝑗𝑝}𝑌^{𝑗1𝑗2⋯𝑗𝑝}={(𝑋⊠_(∗1)𝑍)⊠_(∗2)𝑌}^{𝑚1𝑚2⋯𝑚𝑝}. (4.25)

In other words, the PA1-symmetry is induced from the operations ∗₁,∗₂ to the operations ⊠_(∗1),⊠_(∗2). The next question is if the property of ∗-analyticity is preserved, that is, if 𝑋,𝑌∈𝔗²(𝑊;∗)_(𝜉0) induces (𝑋⊠_(∗)𝑌)∈𝔗²(𝑊;∗)_(𝜉0), and if 𝑋∈𝔗₂(𝑊;∗)_(𝜉0) and 𝑌∈𝔗²(𝑊;∗)_(𝜉0) induce (𝑋⊠^dual_(∗)𝑌)∈𝔗₂(𝑊;∗)_(𝜉0). The next proposition gives a positive answer.

Proposition 4.5. For a pseudoring (ℝ^𝑛;∗), the following symmetries hold: 𝑋,𝑌∈𝔗²(𝑊;∗)_(𝜉0)⇒(𝑋⊠_(∗)𝑌)∈𝔗²(𝑊;∗)_(𝜉0)if(ℝ^𝑛;∗)iscommutativeorthevectorfield𝑌isaconstant,𝑋∈𝔗₂(𝑊;∗)_(𝜉0),𝑌∈𝔗²(𝑊;∗)_(𝜉0)⇒(𝑋⊠^{𝑑𝑢𝑎𝑙}_(∗)𝑌)∈𝔗₂(𝑊;∗)_(𝜉0)if(ℝ^𝑛;∗)iscommutativeorvectorfield𝑌isaconstant. (4.26)(4.27)

Proof. To show (4.26) we have to show that for the product (𝑋⊠_(∗)𝑌)the PCRE hold under the given assumptions, that is, (𝑋⊠_(∗)𝑌)^𝑘𝑙_,𝑚=[𝔇_(∗)(𝑋⊠_(∗)𝑌)]^𝑠𝑙𝐶^𝑘_{(∗)𝑠𝑚}+[𝔇_(∗)(𝑋⊠_(∗)𝑌)]^𝑘𝑠𝐶^𝑙_{(∗)𝑠𝑚} (4.28) with adequate  𝔇_(∗)(𝑋⊠_(∗)𝑌). With 𝑍^𝑘𝑙∶=(𝑋⊠_(∗)𝑌)^𝑘𝑙=𝑋^𝑖𝑗𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞,  we calculate 𝑍^𝑘𝑙_,𝑚=𝑋^𝑖𝑗_,𝑚𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞+𝑋^𝑖𝑗𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞_,𝑚PCRE=[(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑚}+(𝔇_(∗)𝑋)^𝑖𝑠𝐶^𝑗_{(∗)𝑠𝑚}]𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞+𝑋^𝑖𝑗𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}[(𝔇_(∗)𝑌)^𝑠𝑞𝐶^𝑝_{(∗)𝑠𝑚}+(𝔇_(∗)𝑌)^𝑝𝑠𝐶^𝑞_{(∗)𝑠𝑚}]. (4.29) In the case of a constant 𝑌, we obtain 𝑍^𝑘𝑙_,𝑚=[(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑋)^𝑖𝑠𝐶^𝑗_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}]𝑌^𝑝𝑞_{pseudoringsymmetry}=[(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑝}𝐶^𝑘_{(∗)𝑖𝑚}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑋)^𝑖𝑠𝐶^𝑗_{(∗)𝑠𝑞}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑚}]𝑌^𝑝𝑞=[(𝔇_(∗)𝑍)^𝑖𝑙𝐶^𝑘_{(∗)𝑖𝑚}+(𝔇_(∗)𝑍)^𝑘𝑖𝐶^𝑙_{(∗)𝑖𝑚}]𝑌^𝑝𝑞,with(𝔇_(∗)𝑍)^𝑖𝑙∶=(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞, (4.30) and the second statement of (4.26) has been proved. To show the first statement, we start from the partial derivations above. 𝑍^𝑘𝑙_,𝑚=[(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑋)^𝑖𝑠𝐶^𝑗_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}]𝑌^𝑝𝑞+𝑋^𝑖𝑗[(𝔇_(∗)𝑌)^𝑠𝑞𝐶^𝑝_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑌)^𝑝𝑠𝐶^𝑞_{(∗)𝑠𝑚}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}]_{comm.pseudoring}=[(𝔇_(∗)𝑋)^𝑠𝑗𝐶^𝑖_{(∗)𝑠𝑝}𝐶^𝑘_{(∗)𝑖𝑚}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑋)^𝑖𝑠𝐶^𝑗_{(∗)𝑠𝑞}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑚}]𝑌^𝑝𝑞+𝑋^𝑖𝑗[(𝔇_(∗)𝑌)^𝑠𝑞𝐶^𝑝_{(∗)𝑠𝑖}𝐶^𝑘_{(∗)𝑚𝑝}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑌)^𝑝𝑠𝐶^𝑞_{(∗)𝑠𝑗}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑚𝑞}]_{renamingindices}=[(𝔇_(∗)𝑋)^𝑖𝑗𝐶^𝑠_{(∗)𝑖𝑝}𝐶^𝑘_{(∗)𝑠𝑚}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑋)^𝑖𝑗𝐶^𝑠_{(∗)𝑗𝑞}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑠𝑚}]𝑌^𝑝𝑞+𝑋^𝑖𝑗[(𝔇_(∗)𝑌)^𝑝𝑞𝐶^𝑠_{(∗)𝑝𝑖}𝐶^𝑘_{(∗)𝑚𝑠}𝐶^𝑙_{(∗)𝑗𝑞}+(𝔇_(∗)𝑌)^𝑝𝑞𝐶^𝑠_{(∗)𝑞𝑗}𝐶^𝑘_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑚𝑠}]=𝐶^𝑘_{(∗)𝑠𝑚}[(𝔇_(∗)𝑋)^𝑖𝑗𝐶^𝑠_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞+𝑋^𝑖𝑗(𝔇_(∗)𝑌)^𝑝𝑞𝐶^𝑠_{(∗)𝑝𝑖}𝐶^𝑙_{(∗)𝑗𝑞}]+𝐶^𝑙_{(∗)𝑠𝑚}[(𝔇_(∗)𝑋)^𝑖𝑗𝐶^𝑠_{(∗)𝑗𝑞}𝐶^𝑘_{(∗)𝑖𝑝}𝑌^𝑝𝑞+𝑋^𝑖𝑗(𝔇_(∗)𝑌)^𝑝𝑞𝐶^𝑠_{(∗)𝑞𝑗}𝐶^𝑘_{(∗)𝑖𝑝}]=𝐶^𝑘_{(∗)𝑠𝑚}(𝔇_(∗)𝑍)^𝑠𝑙+𝐶^𝑙_{(∗)𝑠𝑚}(𝔇_(∗)𝑋)^𝑘𝑠,with(𝔇_(∗)𝑍)^𝑠𝑙∶=(𝔇_(∗)𝑋)^𝑖𝑗𝐶^𝑠_{(∗)𝑖𝑝}𝐶^𝑙_{(∗)𝑗𝑞}𝑌^𝑝𝑞+𝑋^𝑖𝑗(𝔇_(∗)𝑌)^𝑝𝑞𝐶^𝑠_{(∗)𝑝𝑖}𝐶^𝑙_{(∗)𝑗𝑞}=[(𝔇_(∗)𝑋)⊠_(∗)𝑌+𝑋⊠_(∗)(𝔇_(∗)𝑌)]^𝑠𝑙. (4.31) The proof of the statement (4.27) runs the same lines.

We see that the property of ∗-analyticity generally will not be preserved by the operations ⊠_(∗) and ⊠^dual_(∗), only for simple underlying structures (commutative pseudorings) or for a simple operand (a constant). Thus it makes sense to introduce the concept of weak ∗-analytical fields as ⊠_(∗)-products and ⊠^dual_(∗)-products of ∗-analytical fields, which allows applications in general relativity as well as in system theory, but will not be discussed here for space reason.

In this chapter, a class of Embedding Problems has been analyzed with the result that polynomial 𝑛-dimensional vector fields always can be embedded into polynomial 𝑁-dimensional, ∗-analytic vector fields, 𝑁≥𝑛finite. In the case 𝑁→∞,the results hold for arbitrary smooth vector fields. Furthermore, not only the existence of embedding was shown, but the embedding process was described explicitly. Finally new product operations with interesting properties have been analyzed.

5. Conclusion and Outlook

The paper presents and solves a new type of Embedding Problems which are important for solving nonlinear partial differential equations. The most important results are as follows.

(i)A set of algebras was found (so-called “PA1-chains” or “GAPs”), such that for the corresponding function theories a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.(ii)A wide variety of GAPs can be characterized explicitly by presenting their structure constant tensors explicitly.(iii)GAP-Theory endows new explicit solutions of Einsteins field equations from General Relativity Theory and new explicit solutions of the 𝑛-dimensional Navier-Stokes equations.(iv)A new Embedding Theorem was formulated, which allows a wide generalization of the results above: it was shown explicitly that an arbitrary 𝑛-dimensional smooth vector field always can be embedded into a special 𝑁-Dimensional Smooth Vector Field (𝑁≥𝑛) restricted by PA1-symmetries. In other words, the world of PA1-chains is wide enough to allow embedding for rather general (“smooth”) structures.(v)Some consequences of the Embedding Theorem for the explicit solution of nonlinear partial differential equations have been discussed.(vi)A simple example for the Embedding Theorem has been given.

The following investigations and generalizations would be useful, the construction of more general PA1-chains which would generalize our Embedding Theorem.