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Semigroups with complemented congruence lattices. (English) Zbl 0595.20066
Using the decomposition of a semigroup into its $${\mathcal J}$$-classes, the paper gives a characterization of all globally idempotent semigroups whose lattice of congruences is complemented. Furthermore, an arbitrary semigroup has a complemented congruence lattice if and only if it is an inflation of a semigroup characterized in this way. Thus the general problem of describing all semigroups with complemented congruence lattices is reduced to that of studying the question for the class of simple semigroups.
Reviewer: H.Mitsch

##### MSC:
 20M10 General structure theory for semigroups 20M15 Mappings of semigroups 06B15 Representation theory of lattices
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##### References:
 [1] K.Auinger,On a sublattice of the congruence lattice of a strong semilattice of semigroups (submitted). [2] K.Auinger,Completely regular semigroups whose congruence lattice is complemented (submitted). [3] K.Auinger,Semigroups with Boolean congruence lattice (submitted). · Zbl 0623.20050 [4] L. M. Gluskin,Simple semigroups with zero, Dokl, Akad. Nauk SSSR103 (1955), 5-8. [5] J. Grappy,Demi-groupes dont le treillis des congruences est un treillis compl?ment?, S?minaire Dubreil-Pisot, 1963/64, 21, 01-21. [6] J. Grappy,Sur les demi-groupes admettant un certain type de treillis de congruences, C.R. Acad. Sci. Paris,256 (1963), 2980-2982. · Zbl 0118.03002 [7] T. E. Hall,On the lattice of congruences of a semilattice, J. Austral. Math. Soc.12 (1971), 456-460. · Zbl 0238.06004 [8] H. B. Hamilton,Semilattices whose structure lattice is distributive, Semigroup Forum8 (1974), 245-253. · Zbl 0305.06002 [9] G. Lallement andM. Petrich,Structure d’une classe de demi-groupes r?guliers, J. Math. Pures Appl.48 (1969), 345-397. · Zbl 0211.03903 [10] H. Mitsch,Semigroups and their lattice of congruences, Semigroup Forum26 (1983), 1-63. · Zbl 0513.20047 [11] M. Petrich,Structure of regular semigroups, Cahiers Math?matiques 11, Universit? du Languedoc, Montpellier, 1977. [12] M. Petrich,Inverse Semigroups, Wiley, New York, 1984. · Zbl 0546.20053 [13] B. M. Schein,Homomorphisms and subdirect decompositions of semigroups, Pacific J. Math.17 (1966), 529-547. · Zbl 0197.01603 [14] T. Tamura,Indecomposable completely simple semigroups except groups, Osaka Math. J.8 (1956), 35-42. · Zbl 0070.01803 [15] E. J. Tully, Jr.,Semigroups in which each ideal is a retract, J. Austral. Math. Soc.9 (1969), 239-245. · Zbl 0185.04802
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