Finite arithmetical affine complete algebras having no proper subalgebras
A FACS algebra is a finite arithmetical complete algebra having no proper subalgebras. It is known that a variety \(V\) which is arithmetical of finite type is affine complete if and only if it is generated by a FACS algebra; indeed, if \(\mathbf A\) is a finite functionally complete algebra with no proper subalgebras, then the following hold. (1) \(V ({\mathbf A})\) is arithmetical (and hence affine complete), see the first author [ibid. 29, No. 4, 495-502 (1992; Zbl 0774.08003)]. (2) A polynomial \(f : A^ k \to A\) is a term function of \(\mathbf A\) if and only if \(\sigma f({\mathbf x}) = f(\sigma {\mathbf x})\) for all \({\mathbf x} \in A^ k\) and all \(\sigma \in \text{Aut }{\mathbf A}\). (3) Each automorphism \(\sigma\) of \(\mathbf A\) is a polynomial and is a term function if and only if it is the center of \(\text{Aut }{\mathbf A}\). The main purpose of this paper is to investigate the extent to which these properties persist for non-simple FACS algebras. To explain the results we need the concept of a weakly diagonal algebra. This is a finite algebra \(\mathbf A\) with no proper subalgebras such that any subalgebra of \({\mathbf A} \times {\mathbf A}\) contains the graph of an automorphism of \(\mathbf A\). Various necessary and sufficient conditions for weak diagonality are given and it is proved that each of the following statements implies its successor, but that neither of the implications is reversible. (i) \(\mathbf A\) is weakly diagonal. (ii) The terms of \(\mathbf A\) are the polynomials which are compatible with \(\text{Aut }{\mathbf A}\). (iii) \(V({\mathbf A})\) is arithmetical. The penultimate theorem of the paper, whose statement is too technical easily to be given here, extends (2) and (3) to a wide class of non- simple FACS algebras.
scientific article; zbMATH DE number 638597