A characterization of decidable locally finite varieties

Abstract

We describe the structure of those locally finite varieties whose first order theory is decidable. A variety is a class of universal algebras defined by a set of equations. Such a class is said to be locally finite if every finitely generated member of the class is finite. It turns out that in order for such a variety to have a decidable theory it must decompose into the varietal product of three special kinds of varieties; a strongly Abelian variety; an affine variety; and a discriminator variety.

Generating primitive positive clones

“…We say that a variety V is decidable if and only if there is an algorithm which, given any sentence in the language of V, decides if that sentence holds in every algebra in V. In [14], R. McKenzie and M. Valeriote give the following characterization of decidable locally finite varieties: THEOREM 4.10. [14] Let V be any decidable locally finite variety.…”

“…[14] Let V be any decidable locally finite variety. There exists a strongly Abelian variety V 1 , an affine variety V 2 , and a discriminator variety V 3 so that…”