Assertionally equivalent quasivarieties

A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation tau naturally induces a deductive system, called the tau-assertional logic of K. Two quasivarieties are tau-assertionally equivalent if they have the same tau-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as tau-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation.