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Algebra  2013 

The Depth of Generalized Full Terms and Generalized Full Hypersubstitutions

DOI: 10.1155/2013/396464

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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

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