HSP\(\neq SHPS\) for metabelian groups, and related results
Consider the standard operators on classes of algebras, H (homomorphic images), S (subalgebras) and P (products), and the further operator \(P_ f\) of finite products. It is shown by counterexamples that HSP\(\neq SHPS\) and \(HSP_ f\neq SHP_ fS\) for metabelian groups (groups satisfying \(G''=\{e\})\), and that \(HSP_ f\not\subseteq SHPS\) for solvable groups (in fact, for finite groups satisfying \((G',G'')=\{e\})\). From the first two inequalities and some easier examples, it is deduced that the partially ordered semigroups of operators on metabelian groups generated by \(\{\) H,S,P\(\}\) and by \(\{H,S,P_ f\}\) are as in the ``standard'' 18- element diagram. [Cf. \textit{E. Nelson}, Can. J. Math. 17, 764-768 (1967; Zbl 0149.260), \textit{P. M. Neumann}, Bull. Am. Math. Soc. 76, 1067-1069 (1970; Zbl 0209.321), \textit{D. Pigozzi}, Algebra Univers. 2, 346-353 (1972; Zbl 0272.08006).]
partially ordered semigroups of operators on metabelian groups
scientific article; zbMATH DE number 4019136