Decidable unary varieties (Q1085191)

Let \({\mathcal V}\) denote a variety of unary algebras (in Birkhoff's sense) of finite type which is locally finite, i.e. every finitely generated algebra in \({\mathcal V}\) is finite. Let M(\({\mathcal V})\) be the monoid of all unary terms of \({\mathcal V}\). Then \({\mathcal V}\) has a decidable first order theory if and only if the left ideals of \({\mathcal M}({\mathcal V})\) form an inclusion-ordered chain.