Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties $\mathfrak A_1\subset\mathfrak B_1\subset\mathfrak A_2\subset\mathfrak B_2\subset\dotsb$ such that, for all $i$, an equational theory for $\mathfrak A_i$ and for the class $\mathfrak A_i\cap\mathfrak F$ of all finite semigroups in $\mathfrak A_i$ is undecidable while an equational theory for $\mathfrak B_i$ and for the class $\mathfrak B_i\cap\mathfrak F$ of all finite semigroups in $\mathfrak B_i$ is decidable. An infinite sequence of finitely based semigroup varieties $\mathfrak A_1\supset\mathfrak B_1\supset\mathfrak A_2\supset\mathfrak B_2\supset\dotsb$ is constructed so that, for all $i$, an equational theory for $\mathfrak B_i$ and for the class $\mathfrak B_i\cap\mathfrak F$ of all finite semigroups in $\mathfrak B_i$ is decidable whicle an equational theory for $\mathfrak A_i$ and for the class $\mathfrak A_i\cap\mathfrak F$ of all finite semigroups in $\mathfrak A_i$ is not.