Term minimal algebras
The author develops the counterpart of the theory of minimal algebras from tame congruence theory. Instead of polynomial functions, only term functions are considered, like in a paper of \textit{C. Bergman} and \textit{R. McKenzie} [J. Aust. Math. Soc., Ser. A 48, 133-147 (1990; Zbl 0692.08011)]. In general, one cannot expect such strong results as with polynomials, hence the study is restricted here to strictly simple algebras (i.e., simple algebras without nontrivial proper subalgebras). For example, it remains true for these algebras that their minimal (term) algebras are all isomorphic. For the unary term functions of a term minimal algebra there are four possibilities. Three of these cases have been dealt with earlier by the author [Acta Sci. Math. 51, 251-268 (1987; Zbl 0633.08002); J. Aust. Math. Soc., Ser. A 48, 434-454 (1990; Zbl 0702.08003)] and by the reviewer [Algebra Univers. 18, 262-273 (1984; Zbl 0546.08005)]. The study of the fourth case is the main subject of this paper. In this case the unary term functions form a group \(G\) fixing one element, say 0, and acting regularly on the rest of the elements. Such algebras are called \(G^ 0\)-algebras. The main result establishes that a \(G^ 0\)-algebra is either a 1-dimensional vector space or the idempotent operation \(x\wedge y = 0\), for all \(x \neq y\), is a term operation of the algebra. Then these algebras are classified according to their type in the sense of tame congruence theory. As a consequence, it turns out, for example, that every simple \(G^ 0\)-algebra of type 3 or 4 generates a congruence distributive variety, and for type 3 the variety is congruence 3- permutable as well. For a fixed nontrivial group \(G\), continuum many inequivalent \(G^ 0\)-algebras are constructed. At the end of the paper, very handy figures sketch the structure of the lattice of clones of \(G^ 0\)-algebras.
scientific article; zbMATH DE number 710072
scientific article