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)


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


[1] Balbes, R., Projective and injective distributive lattices, Pacific J. Math., 21, 405-420 (1967) · Zbl 0157.34301
[2] Bergman, G. M., SHPSHSP for metabelian groups, and related results, Algebra Universalis, 26, 267-283 (1989) · Zbl 0627.08006
[3] Bergman, G. M., Partially ordered sets, and minimal systems of counterexamples, Algebra Universalis, 32, 13-30 (1994) · Zbl 0798.06001
[4] Burris, S.; Sankappanavar, H. P., A Course in Universal Algebra (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0478.08001
[5] Comer, S. D.; Johnson, J. S., The standard semigroup of operators of a variety, Algebra Universalis, 2, 77-79 (1972) · Zbl 0266.08003
[6] Grätzer, G., Universal Algebra (1979), Springer-Verlag: Springer-Verlag New York
[7] Grätzer, G.; Lakser, H., The structure of pseudocomplemented distributive lattices II: Congruence extension and amalgamation, Trans. Amer. Math. Soc., 156, 343-358 (1971) · Zbl 0244.06011
[8] Halmos, P. R., Lectures on Boolean Algebras (1963), Van Nostrand: Van Nostrand Princeton · Zbl 0114.01603
[9] Höft, H., A normal form for some semigroups generated by idempotents, Fund. Math., 84, 75-78 (1974) · Zbl 0284.06010
[10] Koppelberg, S., Handbook of Boolean Algebras (1989), North-Holland: North-Holland Amsterdam
[12] McKenzie, R.; McNulty, G. F.; Taylor, W. F., Algebras, Lattices, Varieties. Algebras, Lattices, Varieties, Wadsworth and Brooks/Cole Advanced Books and Software, 1 (1987), Monterey · Zbl 0611.08001
[13] Monk, D., Cardinal Functions on Boolean Algebras (1990), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0706.06009
[14] Nelson, E., Finiteness of semigroups of operators in universal algebra, Canad. J. Math., 19, 764-768 (1967) · Zbl 0149.26004
[15] Nemitz, W.; Whaley, T., Varieties of implicative semilattices, Pacific J. Math., 37, 759-769 (1971) · Zbl 0243.06003
[16] Neumann, P. M., The inequality of SQPS and QSP as operators on classes of groups, Bull. Amer. Math. Soc., 76, 1067-1069 (1970) · Zbl 0209.32102
[17] Pigozzi, D., On some operations on classes of algebras, Notices Amer. Math. Soc., 13, 829 (1966)
[18] Pigozzi, D., On some operations on classes of algebras, Algebra Universalis, 2, 346-353 (1972) · Zbl 0272.08006
[19] Quackenbush, R. W., Structure theory for equational classes generated by quasiprimal algebras, Trans. Amer. Math. Soc., 187, 127-145 (1974) · Zbl 0301.08002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.