×

Varieties of operator semigroups representing partitions. (English) Zbl 0835.20079

A commutative operator semigroup is an algebra with a unary operation and a binary commutative and associative operation. A subpartition of a set is a partition of a subset of the set. The subpartitions on an \(n\)- element set are represented as \(n\)-ary terms in the paper. The authors give necessary and sufficient conditions on the variety ensuring that there are no constant terms and, for each \(n > 0\), that these representing terms be all the essentially \(n\)-ary terms and moreover that distinct subpartitions yield distinct terms. It is shown that there is precisely one such variety of commutative operator semigroup.
Reviewer: J.Duda (Brno)

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
05A18 Partitions of sets
08A40 Operations and polynomials in algebraic structures, primal algebras
08A60 Unary algebras
20M14 Commutative semigroups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] G. Grätzer,Composition of functions, ”Proceedings of the Conference on Universal Algebra”, October 1969 (Kingston, Ontario) (G. H. Wenzel, ed.), Queen’s Papers in Pure and Applied Mathematics, no. 25, Queen’s University, 1970, pp. 1–106.
[2] H. Lakser,A natural representation of partitions as terms of a universal algebra, J. Austral. Math. Soc. Ser. A, (to appear). · Zbl 0797.08004
[3] R. N. McKenzie, G. F. McNulty, and W. F. Taylor, ”Algebras, lattices, varieties, vol. 1”, Wadsworth & Brooks/Cole, Monterey, California, 1987. · Zbl 0611.08001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.