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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 615014, 18 pages
http://dx.doi.org/10.1155/2011/615014
Research Article

The Order of Hypersubstitutions of Type (2,1)

Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Received 30 December 2010; Accepted 27 March 2011

Academic Editor: H. Srivastava

Copyright © 2011 Tawhat Changphas and Wonlop Hemvong. 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

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to 𝑀 -hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an 𝑀 -hyperidentity for a subset 𝑀 of the set of all hypersubstitutions, the variety is called 𝑀 -solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of 𝑀 -solid varieties. In this paper, we study the order of each hypersubstitution of type (2,1), that is, the order of the cyclic subsemigroup of the monoid of all hypersubstitutions of type (2,1) generated by that hypersubstitution.