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.