Equational theories for classes of finite semigroups (Q2714065)

Theorem 1. There exists an infinite sequence of finitely based varieties of semigroups \(A_1\subset B_1\subset A_2\subset B_2\subset\cdots\) such that for all \(i\) the equational theories of \(A_i\) and of the classes \(A_i\cap F\) of all finite semigroups in \(A_i\) are undecidable while the equational theories of \(B_i\) and of the class \(B_i\cap F\) of all finite semigroups in \(B_i\) are decidable.NEWLINENEWLINENEWLINETheorem 2. There exists an infinite sequence of finitely based varieties of semigroups \(A_1\supset B_1\supset A_2\supset B_2\supset\cdots\) such that for all \(i\) the equational theories of \(A_i\) and of the classes \(A_i\cap F\) of all finite semigroups in \(A_i\) are undecidable while the equational theories of \(B_i\) and of the class \(B_i\cap F\) of all finite semigroups in \(B_i\) are decidable.