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Discussiones Mathematicae General Algebra and Applications 25(1) (2005) 89-101
DOI: https://doi.org/10.7151/dmgaa.1093

T-VARIETIES AND CLONES OF T-TERMS

Klaus Denecke University of Potsdam, Institute of Mathematics Am Neuen Palais, 14415 Potsdam, Germany e-mail: kdenecke@rz.uni-potsdam.de

Prakit Jampachon KhonKaen University, Department of Mathematics KhonKaen, 40002 Thailand e-mail: prajam@.kku.ac.th

Abstract

The aim of this paper is to describe how varieties of algebras of type t can be classified by using the form of the terms which build the (defining) identities of the variety. There are several possibilities to do so. In [3], [19], [15] normal identities were considered, i.e. identities which have the form x » x or s » t, where s and t contain at least one operation symbol. This was generalized in [14] to k-normal identities and in [4] to P-compatible identities. More generally, we select a subset T of W_t(X), the set of all terms of type t, and consider identities from T×T. Since any variety can be described by one heterogenous algebra, its clone, we are also interested in the corresponding clone-like structure. Identities of the clone of a variety V correspond to M-hyperidentities for certain monoids M of hypersubstitutions. Therefore we will also investigate these monoids and the corresponding M-hyperidentities.

Keywords: T-quasi constant algebra, T-identity, j-ideal, T-hyperidentity, clone of T-terms.

2000 Mathematics Subject Classification: 08A40, 08A62, 08B05.

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Received 2 May 2005
Revised 20 June 2005