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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 |

### Keywords:

monoid generated by operators on a class of algebras; homomorphic image; subalgebra; direct product; subdirect product; Hasse diagram; partially ordered monoids
Full Text:
DOI

### References:

[1] | Balbes, R., Projective and injective distributive lattices, Pacific J. Math., 21, 405-420 (1967) · Zbl 0157.34301 |

[2] | Bergman, G. M., SHPS≠HSP 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 · Zbl 0412.08001 |

[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 · Zbl 0671.06001 |

[11] | T. R. Lopes, Some semigroups of operators on classes of algebras, unpublished.; T. R. Lopes, Some semigroups of operators on classes of algebras, unpublished. |

[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 |

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