International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 68534, 19 pages
doi:10.1155/IJMMS/2006/68534

Unsolid and fluid strong varieties of partial algebras

S. Busaman1 and K. Denecke2

1Department of Mathematics and Computer Science, Prince of Songkla University, Pattani 94000, Thailand
2Institute of Mathematics, University of Potsdam, Am Neuen Palais, Potsdam 14415, Germany

Received 8 January 2006; Revised 14 August 2006; Accepted 21 August 2006

Copyright © 2006 S. Busaman and K. Denecke. 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

A partial algebra 𝒜=(A;(fiA)i∈I) consists of a set A and an indexed set (fiA)i∈I of partial operations fiA:Ani⊸→A. Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of terms p≈q over the partial algebra 𝒜 is said to be a strong identity in 𝒜 if the right-hand side is defined whenever the left-hand side is defined and vice versa, and both are equal. A strong identity p≈q is called a strong hyperidentity if when the operation symbols occurring in p and q are replaced by terms of the same arity, the identity which arises is satisfied as a strong identity. If every strong identity in a strong variety of partial algebras is satisfied as a strong hyperidentity, the strong variety is called solid. In this paper, we consider the other extreme, the case when the set of all strong identities of a strong variety of partial algebras is invariant only under the identical replacement of operation symbols by terms. This leads to the concepts of unsolid and fluid varieties and some generalizations.