AlgebraVolume 2013 (2013), Article ID 396464, 9 pageshttp://dx.doi.org/10.1155/2013/396464

Research Article

The Depth of Generalized Full Terms and Generalized Full Hypersubstitutions

Sarawut Phuapong and Sorasak Leeratanavalee

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 7 April 2013; Revised 3 June 2013; Accepted 3 June 2013

Academic Editor: Andrei V. Kelarev

Copyright © 2013 Sarawut Phuapong and Sorasak Leeratanavalee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . Such as for a type , 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 -hyperidentities and -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 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 be a countably infinite set of symbols called variables. We often refer to these variables as letters, to as an alphabet, and also refer to the set as an -element alphabet. Let be an indexed set which is disjoint from . Each is called -ary operation symbol, where is a natural number. Let be a function which assigns to every the number as its arity. The function , on the values of written as is called a type.

An -ary term of type, , or simply an -ary term is defined inductively as follows: (i)the variables are -ary terms. (ii)If are -ary terms, then is an -ary term.

By we mean the smallest set which contains and is closed under finite application of (ii). It is clear that every -ary term is also an -ary term for all . The set is the set of all terms of type over the alphabet . This set can be used as the universe of an algebra of type . For every , an -ary operation on is defined by with . The algebra is called the absolutely free algebra of type over the set . The term algebra is generated by the set and has the property called absolute freeness, meaning that for every algebra and every mapping there exists a unique homomorphism which extends the mapping and such that , where is the embedding of into . This can be shown by Figure 1.

Figure 1: The embedding of into .

It is clear that the absolutely free algebras and are isomorphic if and 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 , the depth of a term which is denoted by and is defined as follows: (i)if , then , (ii)if where , then .

A generalized hypersubstitution of type is a mapping which does not necessarily preserve the arity.

We denoted the set of all generalized hypersubstitutions of type by . To define a binary operation on , we need to define the concept of generalized superposition of terms by the following steps.

For any term , (i)if , then , (ii)if , then , (iii)if , then

Then the generalized hypersubstitution can be extended to a mapping by the following steps: (i), (ii), for any -ary operation symbol where , are already defined.

We defined a binary operation on by , where denotes the usual composition of mappings and . Let be the hypersubstitution which maps each -ary operation symbol to the term . In [6], it is proved that for arbitrary terms and for arbitrary generalized hypersubstitutions we have (i), (ii).

It turns out that is a monoid where is its identity element. The monoid of all arity preserving hypersubstitutions of type forms a submonoid of . 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 , or simply a generalized full term, and describe the behavior of the depth of a generalized full term.

Definition 1. Let be the set of operation symbols and with arity : (i)if is a permutation, then is a generalized full term, (ii)if are natural numbers greater than , then is a generalized full term where is a permutation on , (iii)if are generalized full terms, then is a generalized full term. Let be the set of all generalized full terms and let be the set of all permutations on . By Definition 1, the set is closed under finite application of (iii). Therefore is a subalgebra of .

Proposition 2. Let . Then is also a generalized full term.

Proof. We give a proof by induction on the complexity of a generalized full term .If where , then: which is a generalized full term.If where are natural numbers greater than and is a permutation on , then which is a generalized full term.If where and assume that are generalized full terms for all , then By Definition 1(iii), is a generalized full term.

Now we consider the algebra of type with is a permutation on as a generating system. The algebra is called the clone of generalized full terms of type . Then we have the following.

Proposition 3. The algebra satisfies the so-called superassociative law where is an -ary operation symbol and are variables.

Proof. We give a proof by induction on the complexity of the generalized full term which is substituted for . Substituting for a term of the form for , then If we substitute for a term where are natural numbers greater than and is a permutation on , then If we substitute for a term and assume that is satisfied for , then

An algebra of type which satisfies the condition is called a Menger algebra of rank .

Lemma 4. Let . If , then

Proof. Assume that . Then where or where are natural numbers greater than and is a permutation on :if , then If , then

Theorem 5. Let . Then or .

Proof. We prove this fact by induction on the complexity of .If , then by Lemma 4 we have or .If , then we have and assume that the formula is satisfied for that is or for all . Then we have

3. The Depth of Generalized Full Terms with respect to

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

Definition 6. Let and let be a fixed element of and be the set of all variables occurring in . For each variable , the depth of a generalized full term with respect to denoted by is defined inductively as follows: (i)if where , then , (ii)if where are natural numbers greater than and is a permutation on , then , (iii)if and assume that are already defined, then

Theorem 7. Let . Then

Proof. We prove this fact by induction on the complexity of . Suppose that . Then where or where are natural numbers greater than , and is a permutation on .If , then . Therefore, If , then . Therefore, If , then where and assume that the formula is satisfied for . Thus

Example 8. Let be a type; that is, we have only one ternary operation symbol, say . Let where , , and . Then we have , , , and . Thus . Consider .

4. Generalized Full Hypersubstitutions and Substitutions of

In this section, we will generalize the concept of full hypersubstitution. For any generalized full term , we define the generalized full term arising from by mapping all variables corresponding to a permutation inductively by the following steps: (i)if where , then , (ii)if where are natural numbers greater than and is a permutation on , then , (iii)if where , then .

It is clear that is a generalized full term for any generalized full term and for any .

Definition 9. A mapping is called a generalized full hypersubstitution.

By we denote the set of all generalized full hypersubstitutions of type . Every generalized full hypersubstitution can be extended to a map defined on as follows: (i), (ii), (iii).

We define a binary operation on by , where denotes the usual composition of functions. Together with the hypersubstitution defined by , one has a monoid . Then we have the following.

Lemma 10. Let , , and . Then

Proof. We give a proof by induction on the complexity of a generalized full term .If where , then If where are natural numbers greater than and is a permutation on , then If where and assume that the formula is satisfied for , then

Lemma 11. Let and be a generalized full term. Then is also a generalized full term.

Proof. We give a proof by induction on the complexity of a generalized full term .If where , then . Since is a generalized full term, is also a generalized full term.If where are natural numbers greater than and is a permutation on , then . Since is a generalized full term, is also a generalized full term.If where are generalized full terms for all and assume that are generalized full terms for all , then is also a generalized full term.

Lemma 12. forms a submonoid of the monoid of all generalized hypersubstitutions.

Proof. Since is a generalized full term, the identity-generalized hypersubstitution is a generalized full hypersubstitution. Assume that . Since and are generalized full terms, then by Lemma 11 we have that is a generalized full term. Hence and therefore forms a submonoid of .

Definition 13. Let be a generalized full hypersubstitution of type , and then we define

Corollary 14. Let , . Then

Proof. We give a proof by induction on .If , then where or where are natural numbers greater than and is a permutation on .If , then If , then Assume that the for all and . Then

Corollary 15. The function with is a homomorphism from the monoid onto the monoid .

Proof. The mapping is well defined and surjective since for every there exists a generalized full term with . Furthermore, we have and

Proposition 16. Let . Then is an endomorphism on the algebra .

Proof. Indeed is a function. Now we give a proof by induction on the complexity of a term that, for any , First consider the case that where and . Then If where are natural numbers greater than and is a permutation on , then If where and assume that the formula is satisfied for , then

Since the free algebra is generated by the set

therefore any mapping from into can be uniquely extended to an endomorphism of . These mappings are called generalized full clone substitutions. We denote the set of all generalized full clone substitutions by . The set with a binary operation is defined by , where is the usual composition of functions, and, together with , the identity mapping on is a monoid. Then we have the following proposition.

Proposition 17. The monoid can be embedded into the monoid .

Proof. Let . Then by Proposition 16, is an endomorphism on the algebra . Since is a permutation on is a generating set of , the mapping is a substitution with . We define the mapping by . Injectivity of is clear. We show that is a homomorphism. Let . Then . Clearly, the mapping preserves the identity element.

5. Full Strong Hyperidentities and Identities in

Let be the variety of type and let be the set of all identities of consisting of generalized full terms. Then we have the following.

Proposition 18. is a congruence on .

Proof. We will prove that if , , then . Firstly, we give a proof by induction on the complexity of a term that, for every from , there follows .If and , then where is compatible with the operation of the absolutely free algebra and by the definition of generalized full terms.If where are natural numbers greater than and is a permutation on , then Assume now that and, for , we have already Then Now we prove the implication This is a consequence of the fully invariance of as a congruence on the absolutely free algebra and the definition of a generalized full term. Assume now that . Then

Definition 19. Let be a variety of type and let be the set of all identities of consisting of generalized full terms. Then is called an -hyperidentity in if for every . If every identity in is satisfied as an -hyperidentity, the variety is called -solid.

Proposition 20. If is a fully invariant congruence relation on , then the variety is -solid.

Proof. Let and let be a generalized full hypersubstitution. Since by Proposition 16 the extension of is an endomorphism of , we have .

By Proposition 18 we can form a quotient algebra: which belongs to a variety of a Menger algebra of rank . There is the following connection between clone identities and --hyperidentities in .

Proposition 21. Let be the variety of type and let . If is an identity in , then it is an --hyperidentity in .

Proof. Let be an identity in and let . Then the set of all endomorphisms of and with . By the natural mapping we have and this is a valuation mapping with Then This means that is satisfied as an --hyperidentity in .

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.

References

J. Aczél, “Proof of a theorem on distributive type hyperidentities,” Algebra Universalis, vol. 1, no. 1, pp. 1–6, 1971.

V. D. Belousov, “Systems of quasigroups with generalized identities,” Uspekhi Matematicheskikh Nauk, vol. 20, no. 1, pp. 75–146, 1965.

W. D. Neumann, “Mal'cev conditions, spectra and Kronecker product,” Journal of the Australian Mathematical Society A, vol. 25, no. 1, pp. 103–117, 1978.

W. Taylor, “Hyperidentities and hypervarieties,” Aequationes Mathematicae, vol. 23, no. 1, pp. 30–49, 1981.

K. Denecke, D. Lau, R. Pöschel, and D. Schweigert, “Hyperidentities, hyperequational classes and clone congruences,” in Contributions to General Algebra, vol. 7, pp. 97–118, Hölder-Pichler-Tempsky, Vienna, Austria, 1991.

S. Leeratanavalee and K. Denecke, “Generalized hypersubstitutions and strongly solid varieties,” in General Algebra and Applications, Proceedings of the “59 th Workshop on General Algebra”, “15 th Conference for Young Algebraists Potsdam 2000”, pp. 135–145, Shaker, 2000.

K. Denecke, J. Koppitz, and S. Shtrakov, “The depth of a hypersubstitution,” Journal of Automata, Languages and Combinatorics, vol. 6, no. 3, pp. 253–262, 2001.

K. Denecke and Th. Changphas, “Full hypersubstitutions and fully-solid varieties of semigroups,” East-West Journal of Mathematics, vol. 4, no. 1, pp. 101–112, 2002.

A. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, NY, USA, 2003.

A. V. Kelarev, Ring Constructions and Applications, World Scientific, River Edge, NJ, USA, 2002.

J. Koppitz and K. Denecke, M-Solid Varieties of Algebras, Springer Science+Business Media, New York, NY, USA, 2006.