×

zbMATH — the first resource for mathematics

Semigroups with Boolean congruence lattices. (English) Zbl 0623.20050
A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, with Boolean (...) congruence lattice are completely determined.

MSC:
20M10 General structure theory for semigroups
06C15 Complemented lattices, orthocomplemented lattices and posets
06B15 Representation theory of lattices
20M15 Mappings of semigroups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Auinger, K.: Completely regular semigroups whose congruence lattice is complemented. Submitted. · Zbl 0595.20066
[2] Auinger, K.: Semigroups with complemented congruence lattices. Alg. Univ.22, 192-204 (1986). · Zbl 0605.20058 · doi:10.1007/BF01224025
[3] Clifford, A. H., Preston, G. B.: The algebraic theory of semigroups I. Amer. Math. Soc. Surveys No. 7, Providence, R. I.: AMS. 1961. · Zbl 0111.03403
[4] Fountain, J., Lockley, P.: Semilattices of groups with distributive congruence lattice. Semigroup Forum14, 81-91 (1977). · Zbl 0392.20041 · doi:10.1007/BF02194656
[5] Fountain, J., Lockley, P.: Bands with distributive congruence lattice. Proc. Royal Soc. Edinburgh, Sect. A,84, 235-247 (1979). · Zbl 0424.20057
[6] Gr?tzer, G.: General lattice theory. Basel-Stuttgart: Birkh?user. 1978.
[7] Gluskin, L. M.: Simple semigroups with zero. Dokl. Akad. Nauk SSSR103, 5-8 (1955). (Russian).
[8] Gluskin, L. M.: Completely simple semigroups. Har’kov. Gos. Univ. Ucen. Zap.18, 33-39 (1956). (Russian).
[9] Hamilton, H.: Semilattices whose structure lattice is distributive. Semigroup Forum8, 245-253 (1974). · Zbl 0305.06002 · doi:10.1007/BF02194765
[10] Hamilton, H., Nordahl, T.: Semigroups whose lattice of congruences is Boolean. Pacific J. Math.77, 131-143 (1978). · Zbl 0413.20044
[11] Mitsch, H.: Semigroups and their lattices of congruences. Semigroup Forum26, 1-63 (1983). · Zbl 0513.20047 · doi:10.1007/BF02572819
[12] Sali?, V. N.: A compactly generated lattice with unique complements is distributive. Mat. Zametki12, 617-620 (1972) (Russian).
[13] Schein, B. M.: Homomorphisms and subdirect decompositions of semigroups. Pacific J. Math.17, 529-547 (1966). · Zbl 0197.01603
[14] Scott, W. R.: Group theory. Englewood Cliffs, N. J.: Prentice-Hall. 1973.
[15] Tamura, T.: Indecomposable completely simple semigroups except groups. Osaka Math. J.8, 35-42 (1956). · Zbl 0070.01803
[16] Zitomirski?, G. I.: Generalized groups with a Boolean lattice of congruence relations. Uporjadoc Mnozestra i Resetki, Saratov Univ.2, 18-27 (1974). (Russian).
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.