On the partially ordered semigroup generated by the class operators \(I,R,H,S,P\) (Q5947337)

Let \(K\) be a class of similar algebraic structures. Denote by \(I(K)\), \(H(K)\), \(S(K)\) and \(P(K)\), respectively, the classes of all isomorphic images, homomorphic images, algebras isomorphic to a subalgebra, and direct products of algebras in \(K\). Finally, let \(R(K)\) be the class of all algebras isomorphic to a retract of some algebra in \(K\), that is, to a subalgebra \(B\) of some \(A\in K\) such that there is a homomorphism from \(A\) to \(B\) whose restriction to \(B\) is the identity function. With respect to composition ``\(\circ\)'' of operators and the partial ordering: \(O_1\leq O_2\) iff \(O_1(K) \subseteq O_2(K)\), the semigroup generated by \(\{I,H,S,P,R\}\) forms a partially ordered (p.o.) monoid \(M_r\). It is shown that \(M_r\) consists of 25 elements and the multiplication table of \((M_r,\circ)\) and the diagram of \((M_r,\leq)\) are given. (The structure of the p.o. semigroup generated by \(\{H,S,P\}\) was determined by \textit{D. Pigozzi} [Algebra Univers. 2, 346-353 (1972; Zbl 0272.08006)]. Finally, the (full) standard p.o. semigroups are characterized for the varieties of all commutative semigroups, of all monounary algebras, of all abelian groups, and of all groups, respectively.