An approach to tame congruence theory via subtraces
In the present paper the ideas and results of the so-called ``tame congruence theory'', which was developed by \textit{D. Hobby} and \textit{R. McKenzie} in their book ``The structure of finite algebra'' (1988; Zbl 0721.08001), are generalized. The basic ideas of this theory can be outlined as follows: Let \(A\) be a finite algebra. A prime quotient of congruences is a pair \((\alpha,\beta)\) of congruences such that \(\beta\) covers \(\alpha\) in \(\text{Con }A\). So-called tame quotients are considered which are generalizations of prime quotients. There are five types of tame quotients. The type set of \(A\) which is defined as the set of all types of all prime quotients of \(A\) then determines the algebraic properties of \(A\) and of the variety generated by \(A\). In the paper under review so-called subtraces of \(A\) are considered. These are special two- element subsets of \(A\) which have certain properties with respect to the unary polynomials of \(A\). There are five types of subtraces according to the action of the binary polynomials of \(A\) on these subtraces. Now to each prime quotient a certain non-empty set of subtraces can be assigned. It turns out that all subtraces corresponding to the same prime quotient have the same type as this prime quotient. The advantage of the approach via subtraces is that the theory can be generalized to certain infinite, so-called core-finite algebras. Moreover, many of the proofs become simpler. So-called pretame quotients (which are generalizations of tame quotients) are also considered. Finally, a lot of varieties are described all of whose members are core-finite.
scientific article; zbMATH DE number 488264