Total algebras and weak independence. II

Abstract

The theory of primal algebras has recently been generalized in several ways. In particular the concept has been extended on the one hand by the admission of subalgebras and internal isomorphisms to the successively larger classes of semi-primal, infra-primal and quasi-primal algebras [1, 3, 8], and on the other hand by the admission of proper congruences to the hemi-primal algebras [-23. In [53 the authors introduced the concept of "total" algebras: finite algebras in which all mappings which preserve all subalgebras, automorphisms and congruences are representable by polynomials. So defined, this class of algebras was shown to comprehend both the infra-primal and hemi-primal algebras, but, on the other hand, it is easy to see that the quasi-primal algebras (since they may possess non-identical isomorphic subalgebras) are not always total. In the present paper we enlarge the class of total algebras to include the quasi-primals as well as the semi-, infra-, and hemi-primals. Our principal focus is on systems of weakly independent quasi-primal algebras, their products, and the varieties generated by them.