On locally finite varieties with undecidable equational theory

Abstract

In several recent papers [2,3,4,10,14] (and in the doctoral thesis of B. Wells [13]) examples have been found of varieties, V, in which every finitely presented algebra has a decidable word problem but for which no uniform algorithm exists which solves the word problem in an arbitrary algebra from V (the uniform word problem). A second kind of example presented in the above papers are varieties V in which every finitely generated V-free algebra has a decidable word problem but the equational theory of V is undecidable. Varieties with this property are said to be psuedorecursive [14]. A further variation on these ideas are pseudorecursive varieties in which every finitely presented algebra has decidable word problem; we will call varieties with these properties strongly pseudorecursive. Examples of strongly pseudorecursive varieties are also presented in the above papers. It is well known that the undecidability of the equational theory of a variety implies the undecidability of the uniform word problem for that variety. Thus a strongly pseudorecursive variety is also a variety of the first kind described above. Of particular note is the example of Delić [4] which is a finitely based strongly pseudorecursive variety though its basis is very complicated. On the other hand the identity basis for the strongly pseudorecursive variety presented in [3] is infinite but of quite a simple form. The reader is referred to [14] for an interesting discussion on some of the implications of the existence of pseudorecursive varieties.As noted in [14] (see Remark 11.2.4) it is easy to establish the existence of many (strongly) pseudorecursive varieties, even varieties of groups. However all explicitly described examples of strongly pseudorecursive varieties existing in the literature are either non associative varieties of groupoids or varieties of groupoids (sometimes associative) with additional operations. Here we combine a method used in many of the above papers with some existing results to construct comparatively basic strongly pseudorecursive varieties of semigroups (of type 2 ) and a strongly pseudorecursive variety of groups (of type 2, 1, 0 ). Furthermore, rather than presenting particular examples, we describe a large class of examples.

Small semigroup related structures with infinite properties

“…One of the examples presented answers part of a question of Wells in [14]. These results have been submitted in [4].…”

Algorithmic Decidability of Engel’s Property for Automaton Groups

“…The problem of deciding algorithmically whether a group belongs to a given variety has received much attention (see e.g. [17] and references therein); we consider here the harder problems of determining whether a group (respectively a tuple) almost satisfies a given sequence. This has, up to now, been investigated mainly for the torsion sequence above [12].…”