Partially ordered monoids generated by operators \(H,S,P\), and \(H,S,P_f\) are isomorphic (Q5936179)

Given a class \(K\) of algebras of the same type, the filtered product of a family \(A_i\) \((i\in I)\) of algebras in \(K\) with respect to a filter \(F\) over \(I\) is defined as the algebra \(\prod_{i\in I} A_i/\theta_F\) where the congruence \(\theta_F\) is given by: \((a,b)\in\theta_F\) if and only if \(\{i\in I\mid a(i)= b(i)\}\in F\). Let \(H\), \(S\), \(P\), and \(P_f\) be the operators on \(K\) which give, respectively, all homomorphic images, subalgebras, direct products, and filtered products of algebras in \(K\). Denoting by \(M\) and \(M_f\) the partially ordered monoids generated by \(\{H,S,P\}\) and \(\{H,S,P_f\}\), respectively [see \textit{D. Pigozzi}, Algebra Univers. 2, 346-353 (1972; Zbl 0272.08006)], it is shown that \(M\) and \(M_f\) are isomorphic.