The unary case

Abstract

In this chapter we will find necessary and sufficient conditions for a locally finite, strongly Abelian variety V to be decidable. It will turn out that for such varieties, the properties of being undecidable, hereditarily undecidable, unstructured, or ω-unstructured, all coincide. The results of Chapter 11 reduce the problem to determining those locally finite, essentially unary, k-sorted varieties (for k ≥ 1) which are decidable. For the purposes of determining the decidability of V, we may assume that V is in fact a multi-unary k-sorted variety of finite type. Thus each term in the language of V has at most one variable.