The Depth of Generalized Full Terms and Generalized Full Hypersubstitutions

Abstract

We generalize the concept of a full term which was introduced by Denecke and Changphas and the concept of the depth of a hypersubstitution which was introduced by Denecke et al. to the depth of a generalized full term and the depth of a generalized full hypersubstitution and then derive the formular for its depth.

1. Introduction

Identities are used to classify algebras into collections called varieties. An important activity is to classify all varieties of algebras of a given type 𝜏. Such as for a type 𝜏=(2), if the binary operation symbol satisfies the associative law, the algebras are semigroups. Hyperidentities are also used to classify varieties into collections called hypervarieties. Its generalization is called 𝑀-hyperidentities and 𝑀-solid varieties. The study of hyperidentities is a part of Universal Algebra and also a part of Model Theory of second-order language. The notions of hyperidentities and hypervarieties of a given type 𝜏 without nullary operations were originated by Aczél [1], Belousov [2], Neumann [3], and Taylor [4]. The main tool used to study hyperidentities and hypervarieties is the concept of a hypersubstitution which was introduced by Denecke et al. [5]. In 2000, Leeratanavalee and Denecke generalized the concept of hypersubstitution [6].

Let 𝑋={𝑥₁,𝑥₂,𝑥₃,…} be a countably infinite set of symbols called variables. We often refer to these variables as letters, to 𝑋 as an alphabet, and also refer to the set 𝑋_𝑛={𝑥₁,𝑥₂,…,𝑥_𝑛} as an 𝑛-element alphabet. Let (𝑓_𝑖)_𝑖∈𝐼 be an indexed set which is disjoint from 𝑋. Each 𝑓_𝑖 is called 𝑛_𝑖-ary operation symbol, where 𝑛_𝑖≥1 is a natural number. Let 𝜏 be a function which assigns to every 𝑓_𝑖 the number 𝑛_𝑖 as its arity. The function 𝜏, on the values of 𝜏 written as (𝑛_𝑖)_𝑖∈𝐼 is called a type.

An 𝑛-ary term of type, 𝜏, or simply an 𝑛-ary term is defined inductively as follows: (i)the variables 𝑥₁,…,𝑥_𝑛 are 𝑛-ary terms. (ii)If 𝑡₁,…,𝑡_𝑛𝑖 are 𝑛-ary terms, then 𝑓_𝑖(𝑡₁,…,𝑡_𝑛𝑖) is an 𝑛-ary term.

By 𝑊_𝜏(𝑋_𝑛) we mean the smallest set which contains 𝑥₁,…,𝑥_𝑛 and is closed under finite application of (ii). It is clear that every 𝑛-ary term is also an 𝑚-ary term for all 𝑚≥𝑛. The set 𝑊_𝜏(𝑋)∶=⋃^∞_𝑛=1𝑊_𝜏(𝑋_𝑛) is the set of all terms of type 𝜏 over the alphabet 𝑋. This set can be used as the universe of an algebra of type 𝜏. For every 𝑖∈𝐼, an 𝑛_𝑖-ary operation 𝑓_𝑖 on 𝑊_𝜏(𝑋) is defined by 𝑓_𝑖∶𝑊_𝜏(𝑋)^𝑛𝑖→𝑊_𝜏(𝑋) with (𝑡₁,…,𝑡_𝑛𝑖)↦𝑓_𝑖(𝑡₁,…,𝑡_𝑛𝑖). The algebra ℱ_𝜏(𝑋)∶=(𝑊_𝜏(𝑋);(𝑓_𝑖)_𝑖∈𝐼) is called the absolutely free algebra of type 𝜏 over the set 𝑋. The term algebra ℱ_𝜏(𝑋) is generated by the set 𝑋 and has the property called absolute freeness, meaning that for every algebra 𝒜∈Alg(𝜏) and every mapping 𝑓∶𝑋→𝒜 there exists a unique homomorphism 𝑓∶ℱ_𝜏(𝑋)→𝒜 which extends the mapping 𝑓 and such that 𝑓∘𝜑=𝑓, where 𝜑∶𝑋→ℱ_𝜏(𝑋) is the embedding of 𝑋 into ℱ_𝜏(𝑋). This can be shown by Figure 1.

Figure 1 

The embedding of 𝑋 into ℱ𝜏(𝑋).

It is clear that the absolutely free algebras ℱ_𝜏(𝑋) and ℱ_𝜏(𝑌) are isomorphic if 𝑋 and 𝑌 have the same cardinality. One can use trees instead of terms. Trees are connected graphs which have no cycles. It can be used to visualize the structure of computer programs. For applications, we can measure the complexity of a tree. Then the depth of a term (or the height of a tree) is defined. Let 𝑡∈𝑊_𝜏(𝑋), the depth of a term 𝑡 which is denoted by depth(𝑡) and is defined as follows: (i)if 𝑡=𝑥∈𝑋, then depth(𝑡)∶=0, (ii)if 𝑡=𝑓_𝑖(𝑡₁,…,𝑡_𝑛𝑖) where 𝑡₁,…,𝑡_𝑛𝑖∈𝑊_𝜏(𝑋), then depth(𝑡)∶=max{depth(𝑡_𝑗)∣1≤𝑗≤𝑛_𝑖}+1.

A generalized hypersubstitution of type 𝜏 is a mapping 𝜎∶{𝑓_𝑖∣𝑖∈𝐼}→𝑊_𝜏(𝑋) which does not necessarily preserve the arity.

We denoted the set of all generalized hypersubstitutions of type 𝜏 by Hyp_𝐺(𝜏). To define a binary operation on Hyp_𝐺(𝜏), we need to define the concept of generalized superposition of terms 𝑆^𝑚∶𝑊_𝜏(𝑋)^𝑚+1→𝑊_𝜏(𝑋) by the following steps.

For any term 𝑡∈𝑊_𝜏(𝑋), (i)if 𝑡=𝑥_𝑗,1≤𝑗≤𝑚, then 𝑆^𝑚(𝑥_𝑗,𝑡₁,…,𝑡_𝑚)∶=𝑡_𝑗, (ii)if 𝑡=𝑥_𝑗,𝑚<𝑗∈ℕ, then 𝑆^𝑚(𝑥_𝑗,𝑡₁,…,𝑡_𝑚)∶=𝑥_𝑗, (iii)if 𝑡=𝑓_𝑖(𝑠₁,…,𝑠_𝑛𝑖), then 𝑆^𝑚𝑡,𝑡₁,…,𝑡_𝑚∶=𝑓_𝑖𝑆^𝑚𝑠₁,𝑡₁,…,𝑡_𝑚,…,𝑆^𝑚𝑠_𝑛𝑖,𝑡₁,…,𝑡_𝑚. (1)

Then the generalized hypersubstitution 𝜎 can be extended to a mapping 𝜎∶𝑊_𝜏(𝑋)→𝑊_𝜏(𝑋) by the following steps: (i)𝜎[𝑥]∶=𝑥∈𝑋, (ii)𝜎[𝑓_𝑖(𝑡₁,…,𝑡_𝑛𝑖)]∶=𝑆^𝑛𝑖(𝜎(𝑓_𝑖),𝜎[𝑡₁],…,𝜎[𝑡_𝑛𝑖]), for any 𝑛_𝑖-ary operation symbol 𝑓_𝑖 where 𝜎[𝑡_𝑗], 1≤𝑗≤𝑛_𝑖 are already defined.

We defined a binary operation ∘_𝐺 on Hyp_𝐺(𝜏) by 𝜎₁∘_𝐺𝜎₂∶=𝜎₁∘𝜎₂, where ∘ denotes the usual composition of mappings and 𝜎₁,𝜎₂∈Hyp_𝐺(𝜏). Let 𝜎_id be the hypersubstitution which maps each 𝑛_𝑖-ary operation symbol 𝑓_𝑖 to the term 𝑓_𝑖(𝑥₁,…,𝑥_𝑛𝑖). In [6], it is proved that for arbitrary terms 𝑡,𝑡₁,…,𝑡_𝑛∈𝑊_𝜏(𝑋) and for arbitrary generalized hypersubstitutions 𝜎,𝜎₁,𝜎₂ we have (i)𝑆^𝑛(𝜎[𝑡],𝜎[𝑡₁],…,𝜎[𝑡_𝑛])=𝜎[𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)], (ii)(𝜎₁∘𝜎₂)̂=𝜎₁∘𝜎₂.

It turns out that (Hyp_𝐺(𝜏);∘_𝐺,𝜎_id) is a monoid where 𝜎_id is its identity element. The monoid (Hyp(𝜏);∘_ℎ,𝜎_id) of all arity preserving hypersubstitutions of type 𝜏 forms a submonoid of (Hyp_𝐺(𝜏);∘_𝐺,𝜎_id). In 2001, Denecke et al. introduced the concept of the depth of a hypersubstitution [7]. Also in 2002, Denecke and Changpas defined full terms and studied full hypersubstitutions [8]. In this paper we generalized these concepts to the depth of generalized full terms and generalized full hypersubstitutions.

2. Generalized Full Terms

We use standard terminology established in modern algebra; see [9–11]. In this section, we give the definition of a generalized full term of type 𝜏, or simply a generalized full term, and describe the behavior of the depth of a generalized full term.

Definition 1. Let Ω be the set of operation symbols and 𝑓∈Ω with arity (𝑓)=𝑛: (i)if 𝑠∶{1,…,𝑛}→{1,…,𝑛} is a permutation, then 𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) is a generalized full term, (ii)if 𝑗₁,𝑗₂,…,𝑗_𝑛 are natural numbers greater than 𝑛, then 𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) is a generalized full term where 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, (iii)if 𝑡₁,…,𝑡_𝑛 are generalized full terms, then 𝑓(𝑡₁,…,𝑡_𝑛) is a generalized full term.
Let 𝑊^GF_(𝑛)(𝑋) be the set of all generalized full terms and let 𝑃_𝑛 be the set of all permutations on {1,2,…,𝑛}. By Definition 1, the set 𝑊^GF_(𝑛)(𝑋) is closed under finite application of (iii). Therefore (𝑊^GF_(𝑛)(𝑋);(𝑓_𝑖)_𝑖∈𝐼) is a subalgebra of ℱ_𝜏(𝑋).

Proposition 2. Let 𝑝,𝑝₁,𝑝₂,…,𝑝_𝑛∈𝑊^𝐺𝐹_(𝑛)(𝑋). Then 𝑆^𝑛(𝑝,𝑝₁,…𝑝_𝑛) is also a generalized full term.

Proof. We give a proof by induction on the complexity of a generalized full term 𝑝.
If 𝑝=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛, then: 𝑆^𝑛𝑝,𝑝₁,…,𝑝_𝑛=𝑆^𝑛𝑓_𝑖𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑝₁,…,𝑝_𝑛=𝑓𝑝_𝑠(1),…,𝑝_𝑠(𝑛) (2) which is a generalized full term.
If 𝑝=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝑆^𝑛𝑝,𝑝₁,…,𝑝_𝑛=𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑝₁,…,𝑝_𝑛=𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)} (3) which is a generalized full term.
If 𝑝=𝑓(𝑡₁,…,𝑡_𝑛) where 𝑡₁,…,𝑡_𝑛∈𝑊^GF_(𝑛)(𝑋) and assume that 𝑆^𝑛(𝑡_𝑗,𝑝₁,…,𝑝_𝑛) are generalized full terms for all 1≤𝑗≤𝑛, then 𝑆^𝑛𝑝,𝑝₁,…,𝑝_𝑛=𝑆^𝑛𝑓𝑡₁,…,𝑡_𝑛,𝑝₁,…,𝑝_𝑛=𝑓𝑆^𝑛𝑡₁,𝑝₁,…,𝑝_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑝₁,…,𝑝_𝑛. (4) By Definition 1(iii), 𝑆^𝑛(𝑝,𝑝₁,…,𝑝_𝑛) is a generalized full term.

Now we consider the algebra clone_GF(𝑛)∶=(𝑊^GF_(𝑛)(𝑋);𝑆^𝑛) of type (𝑛+1) with ℱ_GS(𝑛)={𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛))∣𝑠∈𝑃_𝑛}∪{𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)})∣𝑗₁,…,𝑗_𝑛>𝑛,𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}} as a generating system. The algebra clone_GF(𝑛) is called the clone of generalized full terms of type (𝑛). Then we have the following.

Proposition 3. The algebra 𝑐𝑙𝑜𝑛𝑒_𝐺𝐹(𝑛) satisfies the so-called superassociative law (𝐶)𝑆^𝑛𝑋₀,𝑆^𝑛𝑌₁,𝑋₁,…,𝑋_𝑛,…,𝑆^𝑛𝑌_𝑛,𝑋₁,…,𝑋_𝑛≈𝑆^𝑛𝑆^𝑛𝑋₀,𝑌₁,…,𝑌_𝑛,𝑋₁,…,𝑋_𝑛, (5) where 𝑆^𝑛 is an (𝑛+1)-ary operation symbol and 𝑋_𝑙,𝑌_𝑗 are variables.

Proof. We give a proof by induction on the complexity of the generalized full term which is substituted for 𝑋₀. Substituting for 𝑋₀ a term of the form 𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) for 𝑠∈𝑃_𝑛, then 𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛𝑆,…,_𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑓𝑆^𝑛𝑥_𝑠(1),𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛𝑆,…,_𝑛𝑥_𝑠(𝑛),𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑓𝑆^𝑛𝑡_𝑠(1),𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑠(𝑛),𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑓𝑡_𝑠(1),…,𝑡_𝑠(𝑛),𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑡₁,…,𝑡_𝑛,𝑠₁,…,𝑠_𝑛. (6) If we substitute for 𝑋₀ a term 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛𝑆,…,_𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑓𝑆^𝑛𝑥_{𝑠′(𝑗1)},𝑡₁,…,𝑡_𝑛𝑆,…,_𝑛𝑥_{𝑠′(𝑗𝑛)},𝑡₁,…,𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑡₁,…,𝑡_𝑛,𝑠₁,…,𝑠_𝑛. (7) If we substitute for 𝑋₀ a term 𝑡=𝑓(𝑟₁,…,𝑟_𝑛) and assume that (𝐶) is satisfied for 𝑟₁,…,𝑟_𝑛∈𝑊^GF_(𝑛)(𝑋), then 𝑆^𝑛𝑓𝑟₁,…,𝑟_𝑛,𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑓𝑆^𝑛𝑟₁,𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛𝑆,…,_𝑛𝑟_𝑛,𝑆^𝑛𝑡₁,𝑠₁,…,𝑠_𝑛,…,𝑆^𝑛𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑓𝑆^𝑛𝑟₁,𝑡₁,…,𝑡_𝑛,…,𝑆^𝑛𝑟_𝑛,𝑡₁,…,𝑡_𝑛,𝑠₁,…,𝑠_𝑛=𝑆^𝑛𝑆^𝑛𝑓𝑟₁,…,𝑟_𝑛,𝑡₁,…,𝑡_𝑛,𝑠₁,…,𝑠_𝑛. (8)

An algebra (𝑀;𝑆^𝑛) of type (𝑛+1) which satisfies the condition (𝐶) is called a Menger algebra of rank 𝑛.

Lemma 4. Let 𝑡,𝑡₁,𝑡₂,…,𝑡_𝑛∈𝑊^𝐺𝐹_(𝑛)(𝑋). If 𝑑𝑒𝑝𝑡ℎ(𝑡)=1, then 𝑑𝑒𝑝𝑡ℎ𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛𝑡=1𝑜𝑟max𝑑𝑒𝑝𝑡ℎ₁,…,𝑑𝑒𝑝𝑡ℎ𝑡_𝑛+1. (9)

Proof. Assume that depth(𝑡)=1. Then 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛 or 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}:
if 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)), then depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=depth𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑡₁,…,𝑡_𝑛=depth𝑓𝑡_𝑠(1),…,𝑡_𝑠(𝑛)=maxdepth𝑡_𝑠(1),…,depth𝑡_𝑠(𝑛)𝑡+1=maxdepth₁,…,depth𝑡_𝑛+1. (10) If 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}), then depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=depth𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥^′_{𝑠(𝑗𝑛)},𝑡₁,…,𝑡_𝑛=depth𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=1. (11)

Theorem 5. Let 𝑡,𝑡₁,𝑡₂,…,𝑡_𝑛∈𝑊^𝐺𝐹_(𝑛)(𝑋). Then 𝑑𝑒𝑝𝑡ℎ(𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛))=1 or max{𝑑𝑒𝑝𝑡ℎ(𝑡₁),…,𝑑𝑒𝑝𝑡ℎ(𝑡_𝑛)}+𝑑𝑒𝑝𝑡ℎ(𝑡).

Proof. We prove this fact by induction on the complexity of depth(𝑡).
If depth(𝑡)=1, then by Lemma 4 we have depth(𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛))=1 or max{depth(𝑡₁),…,depth(𝑡_𝑛)}+depth(𝑡).
If depth(𝑡)>1, then we have 𝑡=𝑓(𝑠₁,…,𝑠_𝑛) and assume that the formula is satisfied for 𝑠₁,…,𝑠_𝑛∈𝑊^GF_(𝑛)(𝑋) that is depth(𝑆^𝑛(𝑠_𝑘,𝑡₁,…,𝑡_𝑛))=1 or max{depth(𝑡₁),…,depth(𝑡_𝑛)}+depth(𝑠_𝑘) for all 1≤𝑘≤𝑛. Then we have depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=depth𝑓𝑆^𝑛𝑠₁,𝑡₁,…,𝑡_𝑛,…,𝑆^𝑛𝑠_𝑛,𝑡₁,…,𝑡_𝑛=maxdepth𝑆^𝑛𝑠₁,𝑡₁…,𝑡_𝑛𝑆,…,depth^𝑛𝑠_𝑛,𝑡₁,…,𝑡_𝑛𝑡+1=maxmaxdepth₁,…,depth𝑡_𝑛+depth𝑠₁𝑡,…,maxdepth₁,…,depth𝑡_𝑛+depth𝑠_𝑛𝑡+1=maxdepth₁,…,depth𝑡_𝑛+maxdepth𝑠₁,…,depth𝑠_𝑛𝑡+1=maxdepth₁,…,depth𝑡_𝑛+depth(𝑡). (12)

3. The Depth of Generalized Full Terms with respect to 𝑘

In this section, we generalize the concept of the depth of a full term 𝑡 with respect to 𝑘 which were studied by Denecke et al. [7] to the depth of a generalized full term 𝑡 with respect to 𝑘.

Definition 6. Let 𝑡∈𝑊^GF_(𝑛)(𝑋) and let 𝑘 be a fixed element of {1,2,…,𝑛} and var(𝑡) be the set of all variables occurring in 𝑡. For each variable 𝑥_𝑘, the depth of a generalized full term 𝑡 with respect to 𝑘 denoted by depth_𝑘(𝑡) is defined inductively as follows: (i)if 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛, then depth_𝑘(𝑡)=1, (ii)if 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,𝑗₂,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then depth_𝑘(𝑡)=0, (iii)if 𝑡=𝑓(𝑡₁,…,𝑡_𝑛) and assume that depth_𝑘(𝑡_𝑗),1≤𝑗≤𝑛 are already defined, then depth_𝑘(𝑡)=0,if𝑥_𝑘∉var(𝑡);maxdepth_𝑘𝑡_𝑗𝑥∣1≤𝑗≤𝑛;_𝑘∈var𝑡_𝑗+1.otherwise (13)

Theorem 7. Let 𝑡,𝑡₁,𝑡₂,…,𝑡_𝑛∈𝑊^𝐺𝐹_(𝑛)(𝑋). Then 𝑑𝑒𝑝𝑡ℎ𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛≥max𝑑𝑒𝑝𝑡ℎ_𝑗(𝑡)+𝑑𝑒𝑝𝑡ℎ𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var(𝑡). (14)

Proof. We prove this fact by induction on the complexity of depth(𝑡). Suppose that depth(𝑡)=1. Then 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛 or 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,𝑗₂,…,𝑗_𝑛 are natural numbers greater than 𝑛, and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}.
If 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)), then 𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)=𝑆^𝑛(𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)),𝑡₁,…,𝑡_𝑛)=𝑓(𝑡_𝑠(1),…,𝑡_𝑠(𝑛)). Therefore, depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=depth𝑓𝑡_𝑠(1),…,𝑡_𝑠(𝑛)=maxdepth𝑡_𝑠(1),…,depth𝑡_𝑠(𝑛)+1,maxdepth_𝑗(𝑡)+depth𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗(𝑡𝑡∈var)=max1+depth₁,…,1+depth𝑡_𝑛=maxdepth𝑡_𝑠(1),…,depth𝑡_𝑠(𝑛)+1. (15) If 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}), then 𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)=𝑆^𝑛(𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}),𝑡₁,…,𝑡_𝑛)=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}). Therefore, depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=depth𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=1,maxdepth_𝑗(𝑡)+depth𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var(𝑡)=0. (16) If depth(𝑡)>1, then 𝑡=𝑓(𝑝₁,…,𝑝_𝑛) where 𝑝₁,…,𝑝_𝑛∈𝑊^GF_(𝑛)(𝑋) and assume that the formula is satisfied for 𝑝₁,…,𝑝_𝑛. Thus 𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=𝑆^𝑛𝑓𝑝₁,…,𝑝_𝑛,𝑡₁,…,𝑡_𝑛=𝑓𝑆^𝑛𝑝₁,𝑡₁,…,𝑡_𝑛,…,𝑆^𝑛𝑝_𝑛,𝑡₁,…,𝑡_𝑛,depth𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛=maxdepth𝑆^𝑛𝑝₁,𝑡₁,…,𝑡_𝑛𝑆,…,depth^𝑛𝑝_𝑛,𝑡₁,…,𝑡_𝑛+1≥maxmaxdepth_𝑗𝑝₁+depth𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var𝑝₁,…,maxdepth_𝑗𝑝_𝑛+depth𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var𝑝_𝑛+1=maxmaxdepth_𝑗𝑝_𝑘∣1≤𝑗≤𝑛,1≤𝑘≤𝑛;𝑥_𝑗∈var𝑝_𝑘𝑡+1+depth_𝑗𝑥∣1≤𝑗≤𝑛,_𝑗∈∪var𝑝_𝑘∣1≤𝑘≤𝑛=maxmaxdepth_𝑗𝑝_𝑘∣1≤𝑗≤𝑛,1≤𝑘≤𝑛;𝑥_𝑗∈var𝑝_𝑘𝑡+1+depth_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var(𝑡)=maxdepth_𝑗(𝑡)+depth𝑡_𝑗∣1≤𝑗≤𝑛;𝑥_𝑗∈var(𝑡). (17)

Example 8. Let 𝜏=(3) be a type; that is, we have only one ternary operation symbol, say 𝑓. Let 𝑡,𝑡₁,𝑡₂∈𝑊^GF₍₃₎(𝑋) where 𝑡=𝑓(𝑓(𝑥₂,𝑥₁,𝑥₃),𝑓(𝑥₇,𝑥₄,𝑥₅),𝑓(𝑥₃,𝑥₂,𝑥₁)), 𝑡₁=𝑓(𝑥₉,𝑥₅,𝑥₄), 𝑡₂=𝑓(𝑓(𝑥₂,𝑥₃,𝑥₁),𝑓(𝑥₂,𝑥₃,𝑥₁),𝑓(𝑥₂,𝑥₃,𝑥₁)) and 𝑡₃=𝑓(𝑥₆,𝑥₄,𝑥₅). Then we have depth(𝑡)=2, depth(𝑡₁)=1, depth(𝑡₂)=2, and depth(𝑡₃)=1. Thus depth(𝑆³(𝑡,𝑡₁,𝑡₂,𝑡₃))=4. Consider max{depth₁(𝑡)+depth(𝑡₁),depth₂(𝑡)+depth(𝑡₂),anddepth₃(𝑡)+depth(𝑡₃)}=max{2+1,2+2,2+1}=4.

4. Generalized Full Hypersubstitutions and Substitutions of clone_GF(𝑛)

In this section, we will generalize the concept of full hypersubstitution. For any generalized full term 𝑡, we define the generalized full term 𝑡_𝑠 arising from 𝑡 by mapping all variables corresponding to a permutation 𝑠∈𝑃_𝑛 inductively by the following steps: (i)if 𝑡=𝑓(𝑥_𝑟(1),…,𝑥_𝑟(𝑛)) where 𝑟∈𝑃_𝑛, then 𝑡_𝑠=𝑓(𝑥_{𝑠(𝑟(1))},…,𝑥_{𝑠(𝑟(𝑛))}), (ii)if 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝑡_𝑠=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}), (iii)if 𝑡=𝑓(𝑡₁,…,𝑡_𝑛) where 𝑡₁,…,𝑡_𝑛∈𝑊^GF_(𝑛)(𝑋), then 𝑡_𝑠=𝑓((𝑡₁)_𝑠,…,(𝑡_𝑛)_𝑠).

It is clear that 𝑡_𝑠 is a generalized full term for any generalized full term 𝑡 and for any 𝑠∈𝑃_𝑛.

Definition 9. A mapping 𝜎∶Ω→𝑊^GF_(𝑛)(𝑋) is called a generalized full hypersubstitution.

By Hyp_GF(𝑛) we denote the set of all generalized full hypersubstitutions of type (𝑛). Every generalized full hypersubstitution 𝜎 can be extended to a map 𝜎 defined on 𝑊^GF_(𝑛)(𝑋) as follows: (i)𝜎[𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛))]∶=𝜎(𝑓)_𝑠, (ii)𝜎[𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)})]∶=𝑆^𝑛(𝜎(𝑓),𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}), (iii)𝜎[𝑓_𝑖(𝑡₁,…,𝑡_𝑛)]∶=𝑆^𝑛(𝜎(𝑓),𝜎[𝑡₁],…,𝜎[𝑡_𝑛]).

We define a binary operation ∘_GF on Hyp_GF(𝑛) by 𝜎₁∘_GF𝜎₂∶=𝜎₁∘𝜎₂, where ∘ denotes the usual composition of functions. Together with the hypersubstitution 𝜎_id defined by 𝜎_id(𝑓)∶=𝑓(𝑥₁,…,𝑥_𝑛), one has a monoid (Hyp_GF(𝑛);∘_GF,𝜎_id). Then we have the following.

Lemma 10. Let 𝑡∈𝑊^𝐺𝐹_(𝑛)(𝑋), 𝑠∈𝑃_𝑛, and 𝜎∈𝐻𝑦𝑝_𝐺𝐹(𝑛). Then 𝑆^𝑛𝑡,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑆^𝑛𝑡_𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛. (18)

Proof. We give a proof by induction on the complexity of a generalized full term 𝑡.
If 𝑡=𝑓(𝑥_𝑟(1),…,𝑥_𝑟(𝑛)) where 𝑟∈𝑃_𝑛, then 𝑆^𝑛𝑡,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑆^𝑛𝑓𝑥_𝑟(1),…,𝑥_𝑟(𝑛),𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑓𝜎𝑡_{𝑠(𝑟(1))},…,𝜎𝑡_{𝑠(𝑟(𝑛))},𝑆^𝑛𝑡_𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑓𝑥_𝑟(1),…,𝑥_{𝑟(𝑛)𝑠},𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑓𝑥_{𝑠(𝑟(1))},…,𝑥_{𝑠(𝑟(𝑛))},𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑓𝜎𝑡_{𝑠(𝑟(1))},…,𝜎𝑡_{𝑠(𝑟(𝑛))}. (19) If 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝑆^𝑛𝑡,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑆^𝑛𝑡_𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)𝑠},𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}. (20) If 𝑡=𝑓(𝑡^′₁,…,𝑡^′_𝑛) where 𝑡^′₁,…,𝑡^′_𝑛∈𝑊^GF_(𝑛)(𝑋) and assume that the formula is satisfied for 𝑡^′₁,…,𝑡^′_𝑛, then 𝑆^𝑛𝑡,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑆^𝑛𝑓𝑡^′₁,…,𝑡^′_𝑛,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑓𝑆^𝑛𝑡^′₁,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)𝑆,…,_𝑛𝑡^′_𝑛,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑓𝑆^𝑛𝑡^′_1𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛𝑆,…,_𝑛𝑡^′_𝑛𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑓𝑡^′_1𝑠,…,𝑡^′_𝑛𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑓𝑡^′₁,…,𝑡^′_𝑛𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑡_𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛. (21)

Lemma 11. Let 𝜎∈𝐻𝑦𝑝_𝐺𝐹(𝑛) and 𝑝 be a generalized full term. Then 𝜎[𝑝] is also a generalized full term.

Proof. We give a proof by induction on the complexity of a generalized full term 𝑝.
If 𝑝=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛, then 𝜎[𝑝]=𝜎(𝑓)_𝑠. Since 𝜎(𝑓)_𝑠 is a generalized full term, 𝜎[𝑝] is also a generalized full term.
If 𝑝=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝜎[𝑝]=𝜎[𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)})]=𝑆^𝑛(𝜎(𝑓),𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}). Since 𝑆^𝑛(𝜎(𝑓),𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) is a generalized full term, 𝜎[𝑝] is also a generalized full term.
If 𝑝=𝑓(𝑝₁,…,𝑝_𝑛) where 𝑝_𝑗 are generalized full terms for all 1≤𝑗≤𝑛 and assume that 𝜎[𝑝_𝑗] are generalized full terms for all 1≤𝑗≤𝑛, then 𝜎[𝑝]=𝑆^𝑛(𝜎(𝑓_𝑖),𝜎[𝑝₁],…,𝜎[𝑝_𝑛]) is also a generalized full term.

Lemma 12. 𝐻𝑦𝑝_𝐺𝐹(𝑛) forms a submonoid of the monoid 𝐻𝑦𝑝_𝐺(𝑛) of all generalized hypersubstitutions.

Proof. Since 𝜎_id(𝑓)=𝑓(𝑥₁,…,𝑥_𝑛) is a generalized full term, the identity-generalized hypersubstitution is a generalized full hypersubstitution.
Assume that 𝜎₁,𝜎₂∈Hyp_GF(𝑛). Since (𝜎₁∘_GF𝜎₂)(𝑓)=𝜎₁[𝜎₂(𝑓)] and 𝜎₁(𝑓),𝜎₂(𝑓) are generalized full terms, then by Lemma 11 we have that (𝜎₁∘_GF𝜎₂)(𝑓) is a generalized full term. Hence 𝜎₁∘_GF𝜎₂∈Hyp_GF(𝑛) and therefore Hyp_GF(𝑛) forms a submonoid of Hyp_𝐺(𝑛).

Definition 13. Let 𝜎 be a generalized full hypersubstitution of type (𝑛), and then we define depth(𝜎)=maxdepth𝜎𝑓∣𝑓∈Ω. (22)

Corollary 14. Let 𝑝=𝑓(𝑝₁,…,𝑝_𝑛)∈𝑊^𝐺𝐹_(𝑛)(𝑋), 𝜎∈𝐻𝑦𝑝_𝐺𝐹(𝑛). Then 𝑑𝑒𝑝𝑡ℎ𝜎𝑝=𝑑𝑒𝑝𝑡ℎ𝜎𝑓𝑑𝑒𝑝𝑡ℎ𝑝. (23)

Proof. We give a proof by induction on depth(𝑝).
If depth(𝑝)=1, then 𝑝=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑠∈𝑃_𝑛 or 𝑝=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}.
If 𝑝=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)), then depth𝜎𝑝=depth𝜎𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛)=depth𝜎𝑓_𝑠𝜎𝑓𝜎𝑓𝑝.=depth=depthdepth (24) If 𝑝=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}), then 𝜎𝑝𝜎𝑓𝑥depth=depth_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=depth𝑆^𝑛𝜎𝑓,𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=maxdepth𝑥_{𝑠′(𝑗1)},…,depth𝑥_{𝑠′(𝑗𝑛)}𝜎𝑓𝜎𝑓𝜎𝑓𝜎𝑓𝑝.+depth=0+depth=depth=depthdepth (25) Assume that the depth(𝜎[𝑝_𝑘])=depth(𝜎(𝑓))depth(𝑝_𝑘) for all 1≤𝑘≤𝑛 and 𝑝=𝑓(𝑝₁,…,𝑝_𝑛). Then 𝜎𝑝𝑆depth=depth^𝑛𝜎𝑓,𝜎𝑝₁,…,𝜎𝑝_𝑛=depth𝜎𝑓+maxdepth𝜎𝑝_𝑘𝜎𝑓𝜎𝑓𝑝∣1≤𝑘≤𝑛=depth+maxdepthdepth_𝑘𝜎𝑓𝜎𝑓𝑝∣1≤𝑘≤𝑛=depth+depthmaxdepth_𝑘𝜎𝑓𝑝∣1≤𝑘≤𝑛=depth1+maxdepth_𝑘𝜎𝑓𝑝.∣1≤𝑘≤𝑛=depthdepth (26)

Corollary 15. The function 𝑑𝑒𝑝𝑡ℎ^𝐺𝐹∶𝐻𝑦𝑝_𝐺𝐹(𝑛)→ℕ⁺ with 𝜎↦𝑑𝑒𝑝𝑡ℎ(𝜎) is a homomorphism from the monoid (𝐻𝑦𝑝_𝐺𝐹(𝑛);∘_𝐺𝐹,𝜎_𝑖𝑑) onto the monoid (ℕ⁺;⋅,1).

Proof. The mapping depth^GF is well defined and surjective since for every 𝑛∈ℕ⁺ there exists a generalized full term 𝑝 with depth(𝑝)=𝑛. Furthermore, we have depth^GF(𝜎_id)=depth(𝜎_id(𝑓))=depth(𝑓(𝑥₁,…,𝑥_𝑛))=1 and depth𝜎₁∘_GF𝜎₂𝑓=depth𝜎₁𝜎₂𝑓=depth𝜎₁𝑓depth𝜎₂𝑓=depth^GF𝜎₁depth^GF𝜎₂. (27)

Proposition 16. Let 𝜎∈𝐻𝑦𝑝_𝐺𝐹(𝑛). Then 𝜎 is an endomorphism on the algebra (𝑊^𝐺𝐹_(𝑛)(𝑋);𝑆^𝑛).

Proof. Indeed 𝜎∶(𝑊^GF_(𝑛)(𝑋);𝑆^𝑛)→(𝑊^GF_(𝑛)(𝑋);𝑆^𝑛) is a function. Now we give a proof by induction on the complexity of a term 𝑡₀ that, for any 𝑡₁,…,𝑡_𝑛∈𝑊^GF_(𝑛)(𝑋), 𝜎𝑆^𝑛𝑡₀,𝑡₁,…,𝑡_𝑛=𝑆^𝑛𝜎𝑡₀,…,𝜎𝑡_𝑛. (28) First consider the case that 𝑡₀=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) where 𝑖∈𝐼 and 𝑠∈𝑃_𝑛. Then 𝜎𝑆^𝑛𝑡₀,𝑡₁,…,𝑡_𝑛=𝜎𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑡₁,…,𝑡_𝑛=𝜎𝑓𝑡_𝑠(1),…,𝑡_𝑠(𝑛)=𝑆^𝑛𝜎𝑓,𝜎𝑡_𝑠(1),…,𝜎𝑡_𝑠(𝑛)=𝑆^𝑛𝜎𝑓_𝑠,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝜎𝑡₀,𝜎𝑡₁,…,𝜎𝑡_𝑛. (29) If 𝑡₀=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝜎𝑆^𝑛𝑡₀,𝑡₁,…,𝑡_𝑛=𝜎𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑡₁,…,𝑡_𝑛=𝜎𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=𝑆^𝑛𝜎𝑓,𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=𝑆^𝑛𝜎𝑓,𝜎𝑆^𝑛𝑥_{𝑠′(𝑗1)},𝑡₁,…,𝑡_𝑛𝜎𝑆,…,_𝑛𝑥_{𝑠′(𝑗𝑛)},𝑡₁,…,𝑡_𝑛=𝑆^𝑛𝑆^𝑛𝜎𝑓,𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝜎𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝜎𝑡₀,𝜎𝑡₁,…,𝜎𝑡_𝑛. (30) If 𝑡₀=𝑓(𝑟₁,…,𝑟_𝑛) where 𝑟₁,…,𝑟_𝑛∈𝑊^GF_(𝑛)(𝑋) and assume that the formula is satisfied for 𝑟₁,…,𝑟_𝑛, then 𝜎𝑆^𝑛𝑡₀,𝑡₁,…,𝑡_𝑛=𝜎𝑆^𝑛𝑓𝑟₁,…,𝑟_𝑛,𝑡₁,…,𝑡_𝑛=𝜎𝑓𝑆^𝑛𝑟₁,𝑡₁,…,𝑡_𝑛,…,𝑆^𝑛𝑟_𝑛,𝑡₁,…,𝑡_𝑛=𝑆^𝑛𝜎𝑓,𝜎𝑆^𝑛𝑟₁,𝑡₁,…,𝑡_𝑛𝜎𝑆,…,_𝑛𝑟_𝑛,𝑡₁,…,𝑡_𝑛=𝑆^𝑛𝜎𝑓,𝑆^𝑛𝜎𝑟₁,𝜎𝑡₁,…,𝜎𝑡_𝑛𝑆,…,_𝑛𝜎𝑟_𝑛,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝑆^𝑛𝜎𝑓,𝜎𝑟₁,…,𝜎𝑟_𝑛,𝜎𝑡₁,…,𝜎𝑡_𝑛=𝑆^𝑛𝜎𝑡₀,𝜎𝑡₁,…,𝜎𝑡_𝑛. (31)

Since the free algebra clone_GF(𝑛) is generated by the set 𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛)𝑠∈𝑃_𝑛∪𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}∣𝑗₁,…,𝑗_𝑛>𝑛,𝑠^′isapermutationon𝑗₁,…,𝑗_𝑛, (32)

therefore any mapping 𝜂 from ℱ_GS(𝑛) into 𝑊^GF_(𝑛)(𝑋) can be uniquely extended to an endomorphism 𝜂 of clone_GF(𝑛). These mappings are called generalized full clone substitutions. We denote the set of all generalized full clone substitutions by Subst_GFC. The set Subst_GFC with a binary operation ⊙ is defined by 𝜂₁⊙𝜂₂∶=𝜂₁∘𝜂₂, where ∘ is the usual composition of functions, and, together with id_ℱGS(𝑛), the identity mapping on ℱ_GS(𝑛) is a monoid. Then we have the following proposition.

Proposition 17. The monoid (𝐻𝑦𝑝_𝐺𝐹(𝑛);∘_𝐺𝐹,𝜎_𝑖𝑑) can be embedded into the monoid (𝑆𝑢𝑏𝑠𝑡_𝐺𝐹𝐶;⊙,𝑖𝑑_{ℱ𝐺𝑆(𝑛)}).

Proof. Let 𝜎∈Hyp_GF(𝑛). Then by Proposition 16, 𝜎 is an endomorphism on the algebra clone_GF(𝑛). Since ℱ_GS(𝑛)={𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛))∣𝑠∈𝑃_𝑛}∪{𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)})∣𝑗₁,…,𝑗_𝑛>𝑛,𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}} is a generating set of clone_GF(𝑛), the mapping 𝜎_{∣ℱGS(𝑛)} is a substitution with 𝜎_{∣ℱGS(𝑛)}=𝜎. We define the mapping 𝜓∶Hyp_GF(𝑛)→Subst_GFC by 𝜓(𝜎)=𝜎_{∣ℱGS(𝑛)}. Injectivity of 𝜓 is clear. We show that 𝜓 is a homomorphism. Let 𝜎₁,𝜎₂∈Hyp_GF(𝑛). Then 𝜓(𝜎₁∘_GF𝜎₂)=(𝜎₁∘𝜎₂)∣_ℱGS(𝑛)=(𝜎₁∘𝜎₂)_{∣ℱGS(𝑛)}=𝜎₁∘𝜎_{2∣ℱGS(𝑛)}=𝜎_{1∣ℱGS(𝑛)}∘𝜎_{2∣ℱGS(𝑛)}=𝜓(𝜎₁)∘𝜓(𝜎₂)=𝜓(𝜎₁)⊙𝜓(𝜎₂). Clearly, the mapping 𝜓 preserves the identity element.

5. Full Strong Hyperidentities and Identities in clone_GF(𝑛)

Let 𝑉 be the variety of type (𝑛) and let Id^GF𝑉∶=𝑊^GF_(𝑛)(𝑋)²∩Id𝑉 be the set of all identities of 𝑉 consisting of generalized full terms. Then we have the following.

Proposition 18. 𝐼𝑑^𝐺𝐹𝑉 is a congruence on 𝑐𝑙𝑜𝑛𝑒_𝐺𝐹(𝑛).

Proof. We will prove that if 𝑟≈𝑡, 𝑟_𝑘≈𝑡_𝑘∈Id^GF𝑉,𝑘=1,2,…,𝑛, then 𝑆^𝑛(𝑟,𝑟₁,…,𝑟_𝑛)≈𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)∈Id^GF𝑉. Firstly, we give a proof by induction on the complexity of a term 𝑡∈𝑊^GF_(𝑛)(𝑋) that, for every 𝑛∈ℕ⁺ from 𝑡_𝑘≈𝑟_𝑘∈Id^GF𝑉,𝑘=1,2,…,𝑛, there follows 𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)≈𝑆^𝑛(𝑡,𝑟₁,…,𝑟_𝑛)∈Id^GF𝑉.
If 𝑡=𝑓(𝑥_𝑠(1),…,𝑥_𝑠(𝑛)) and 𝑠∈𝑃_𝑛, then 𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑡₁,…,𝑡_𝑛=𝑓𝑡_𝑠(1),…,𝑡_𝑠(𝑛)≈𝑓𝑟_𝑠(1),…,𝑟_𝑠(𝑛)=𝑆^𝑛𝑓𝑥_𝑠(1),…,𝑥_𝑠(𝑛),𝑟₁,…,𝑟_𝑛∈Id^GF𝑉, (33) where Id𝑉 is compatible with the operation 𝑓 of the absolutely free algebra ℱ_(𝑛)(𝑋) and by the definition of generalized full terms.
If 𝑡=𝑓(𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}) where 𝑗₁,…,𝑗_𝑛 are natural numbers greater than 𝑛 and 𝑠^′ is a permutation on {𝑗₁,…,𝑗_𝑛}, then 𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑡₁,…,𝑡_𝑛=𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)}=𝑆^𝑛𝑓𝑥_{𝑠′(𝑗1)},…,𝑥_{𝑠′(𝑗𝑛)},𝑟₁,…,𝑟_𝑛∈Id^GF𝑉. (34) Assume now that 𝑡=𝑓(𝑙₁,…,𝑙_𝑛)∈𝑊^GF_(𝑛)(𝑋) and, for 𝑙_𝑗,1≤𝑗≤𝑛, we have already 𝑆^𝑛𝑙_𝑗,𝑡₁,…,𝑡_𝑛≈𝑆^𝑛𝑙_𝑗,𝑟₁,…,𝑟_𝑛∈Id^GF𝑉. (35) Then 𝑆^𝑛𝑓𝑙₁,…,𝑙_𝑛,𝑡₁,…,𝑡_𝑛=𝑓𝑆^𝑛𝑙₁,𝑡₁,…,𝑡_𝑛,…,𝑆^𝑛𝑙_𝑛,𝑡₁,…,𝑡_𝑛≈𝑓𝑆^𝑛𝑙₁,𝑟₁,…,𝑟_𝑛,…,𝑆^𝑛𝑙_𝑛,𝑟₁,…,𝑟_𝑛=𝑆^𝑛𝑓𝑙₁,…,𝑙_𝑛,𝑟₁,…,𝑟_𝑛∈Id^GF𝑉. (36) Now we prove the implication 𝑡≈𝑟∈Id^GFV⟹𝑆^𝑛𝑡,𝑟₁,…,𝑟_𝑛≈𝑆^𝑛𝑟,𝑟₁,…,𝑟_𝑛∈Id^GF𝑉. (37) This is a consequence of the fully invariance of Id𝑉 as a congruence on the absolutely free algebra ℱ_(𝑛)(𝑋) and the definition of a generalized full term. Assume now that 𝑡≈𝑟,𝑡_𝑘≈𝑟_𝑘∈Id^GF𝑉. Then 𝑆^𝑛𝑡,𝑡₁,…,𝑡_𝑛≈𝑆^𝑛𝑟,𝑡₁,…,𝑡_𝑛≈𝑆^𝑛𝑟,𝑟₁,…,𝑟_𝑛∈Id^GF𝑉. (38)

Definition 19. Let 𝑉 be a variety of type (𝑛) and let Id^GF𝑉 be the set of all identities of 𝑉 consisting of generalized full terms. Then 𝑠≈𝑡∈Id^GF𝑉 is called an Ω-GF-hyperidentity in 𝑉 if 𝜎[𝑠]≈𝜎[𝑡]∈Id^GF𝑉 for every 𝜎∈Hyp_GF(𝑛). If every identity in Id^GF𝑉 is satisfied as an Ω-GF-hyperidentity, the variety 𝑉 is called Ω-GF-solid.

Proposition 20. If 𝐼𝑑^𝐺𝐹𝑉 is a fully invariant congruence relation on 𝑐𝑙𝑜𝑛𝑒_𝐺𝐹(𝑛), then the variety 𝑉 is Ω-𝐺𝐹-solid.

Proof. Let 𝑠≈𝑡∈Id^GF𝑉 and let 𝜎∈Hyp_GF(𝑛) be a generalized full hypersubstitution. Since by Proposition 16 the extension 𝜎 of 𝜎 is an endomorphism of clone_GF(𝑛), we have 𝜎[𝑠]≈𝜎[𝑡]∈Id^GF𝑉.

By Proposition 18 we can form a quotient algebra: clone_GF𝑉∶=clone_GF(𝑛)/Id^GF𝑉 (39) which belongs to a variety of a Menger algebra of rank 𝑛. There is the following connection between clone identities and Ω-GF-hyperidentities in 𝑉.

Proposition 21. Let 𝑉 be the variety of type (𝑛) and let 𝑠≈𝑡∈𝐼𝑑^𝐺𝐹𝑉. If 𝑠≈𝑡 is an identity in 𝑐𝑙𝑜𝑛𝑒_𝐺𝐹(𝑛), then it is an Ω-𝐺𝐹-hyperidentity in 𝑉.

Proof. Let 𝑠≈𝑡∈Id^GF𝑉 be an identity in clone_GF(𝑛) and let 𝜎∈Hyp_GF(𝑛). Then 𝜎∈End(clone_GF𝜏) the set of all endomorphisms of clone_GF(𝑛) and 𝜎_∣ℱGS(n)∈Subst_GFC with 𝜎_{∣ℱGS(𝑛)}=𝜎. By the natural mapping natId^GF𝑉 we have natId^GF𝑉∘𝜎_{∣ℱGS(𝑛)}∶ℱ_GS(𝑛)⟶clone_GF(𝑛), (40) and this is a valuation mapping with natId^GF𝑉∘𝜎_{∣ℱGS(𝑛)}=natId^GF𝑉∘𝜎. (41) Then 𝑠≈𝑡∈Idclone_GF𝑉⟹natId^GF𝑉∘𝜎_{∣ℱGS(𝑛)}(𝑠)=natId^GF𝑉∘𝜎_{∣ℱGS(𝑛)}(𝑡)⟹natId^GF𝑉∘𝜎(𝑠)=natId^GF𝜎(𝑡)⟹[𝜎[𝑠𝑉∘]]_Id^GF_𝑉=[𝜎[𝑡]]_Id^GF_𝑉⟹𝜎[𝑠]≈𝜎[𝑡]∈Id^GF𝑉. (42) This means that 𝑠≈𝑡 is satisfied as an Ω-GF-hyperidentity in 𝑉.

Acknowledgments

This work was supported by the Higher Education Commission, and the authors were supported by CHE Ph. D. Scholarship and the Faculty of Science of Chiang Mai University, Thailand. The authors are grateful to four referees for thorough reports that have helped us to improve the paper.