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On the partially ordered monoid generated by the operators \(H, S, P, P_s\) on classes of algebras. (English) Zbl 0999.08006

For a class \(K\) of algebras of the same type let \(H(K)\), \(S(K)\), \(P(K)\), \(P_s(K)\) denote the classes of all algebras isomorphic to a homomorphic image, subalgebra, direct product, subdirect product of algebras in \(K\), respectively. In this paper the monoid \(M_s\) generated by the operators \(H\), \(S\), \(P\), \(P_s\) with respect to the compositon \((X\circ Y)(K)= X(Y(K))\) is described. It is shown that \(M_s\) has 22 elements, i.e., that there are 22 different operators such that every composition of \(H\), \(S\), \(P\) and \(P_s\) coincides with one of them. Also the Hasse diagram of the set \(M_s\) partially ordered by \(X\leq Y\) iff \(X(K)\subseteq Y(K)\) is given thus describing the positively ordered monoid \((M_s,\circ,\leq)\) completely. Restricting the domain of the operators \(H\), \(S\), \(P\) and \(P_s\) to the class \(K\) of all groups, Abelian groups, unary algebras (of countable unary type), Boolean algebras and distributive lattices, respectively, it is proved that \(M_s(K)\) has 22, 17, 17, 12 and 12 elements, respectively. Also the corresponding Hasse diagrams are provided. The problems concerning the partially ordered monoids \(M_r\) and \(M_f\) generated by the operators \(H\), \(S\), \(P\), \(R\) (retracts) respectively by \(H\), \(S\), \(P_f\) (filtered products) was dealt with by the author in: Order 18, 49-60 (2001; Zbl 0990.08005) and in Semigr. Forum 62, 485-490 (2001; Zbl 0984.08008), respectively.
Reviewer: H.Mitsch (Wien)

MSC:

08C99 Other classes of algebras
06F05 Ordered semigroups and monoids
08B25 Products, amalgamated products, and other kinds of limits and colimits
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