×

Semigroups with \(\wedge\)-semidistributive subsemigroup lattices. (English) Zbl 0549.20040

Initiated largely by L. N. Shevrin, the study of the subsemigroup lattices of semigroups has been extensive (see [L. N. Shevrin and A. J. Ovsyannikov, Semigroup Forum 27, 1-154 (1983; Zbl 0523.20037)]). Among the first results was a description of those semigroups S whose subsemigroup lattices L(S) are distributive. Subsequently, descriptions of those semigroups S for which L(S) satisfies various other lattice-theoretic properties have been given. In this paper the author describes those semigroups S for which L(S) has the property in the title, that is, satisfying the implication \((SD_{\wedge})\) introduced by Jónsson in the study of free lattices: \(a\wedge b=a\wedge c\Rightarrow a\wedge (b\vee c)=a\wedge b.\) As for distributivity, the description, though complete, is too complicated to be reproduced here. The author’s other main result is that the subgroup lattice of a group is \(\wedge\)-semidistributive if and only if it is distributive.
Reviewer: P.R.Jones

MSC:

20M10 General structure theory for semigroups
20E15 Chains and lattices of subgroups, subnormal subgroups
06D05 Structure and representation theory of distributive lattices
08A30 Subalgebras, congruence relations

Citations:

Zbl 0523.20037
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Benzaken, C. and H.C. Mayr,Notion de demi-bande.Demi-bandes de type deux, Semigroup Forum, 10 (1975), 115–128. · Zbl 0303.20041
[2] Clifford, A.H. and G.B. Preston,The Algebraic Theory of Semigroups, Amer. Math. Soc., Providence, R. I., I (1961). · Zbl 0111.03403
[3] Cohn, P.,Universal Algebra, Harper & Row, New York, Evanston and London (1965).
[4] Curzio, M,Alcune osservazione sul reticolo dei sottogruppi d’un gruppo finito. Ricerche mat. 6 (1957), 96–110. · Zbl 0079.25402
[5] Ego, M.,Structure des demi-groupes dont le treillis des sous-demi-groupes est distributif, C. r. Acad. Sci. 252 (1961), 2490–2492. · Zbl 0112.25601
[6] Ego, M.,Structure des demi-groupes dont le treillis des sous-demi-groupes est modulaire ou semi-modulaire, C. r. Acad. Sci. 254 (1962), 1723–1725. · Zbl 0114.01801
[7] Gierz, G., K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott,A Compendium of Continuous Lattices, Springer-Verlag, Berlin, Heidelberg, New-York (1980). · Zbl 0452.06001
[8] Gratzer, G.,General Lattice Theory, Akademie-Verlag, Berlin (1978).
[9] Igoshin, V.I.,On lattices of algebraic systems quasivarieties, Ordered sets and lattices, 5 Saratov (1978), 44–55 (in Russian). · Zbl 0402.08016
[10] Kurosh, A.G.,Group Theory, Nauka, Moscow (1967) (in Russian).
[11] Napolitani, F.,Elementi quasidistributivi nel reticolo dei sottogruppi di un gruppo, Ricerche mat. 14 (1965), 93–101. · Zbl 0133.28301
[12] Napolitani, F.,Elementi -quasidistributivi ed u.c.r.elementi del reticolo dei sottogruppi di un gruppo finito, Ricerche mat. 17 (1968), 95–108. · Zbl 0195.31802
[13] Papert, D.,Congruences in semilattices, J. of The London Math. Soc. 39 (1964), 723–729. · Zbl 0126.03802
[14] Petrich, M.,Lectures in Semigroups, Akademie-Verlag, Berlin (1977). · Zbl 0369.20036
[15] Prosvirov, A.S.,On periodic semigroups, Mat. Zap. Ural. Univ., Sverdlovsk, 8 (1971), 77–94.
[16] Schmidt, J.,Binomial pairs, semi-Browerian and Browerian semilattices, Notre Dame J. of Formal Logic, 19 (1978), 421–434. · Zbl 0373.02043
[17] Shevrin, L.N.,Semigroups with certain classes of subsemigroup lattices, Dokl. Akad. Nauk SSSR, 138 (1961), 796–798 (in Russian).
[18] Shevrin, L.N.,On lattice properties of semi-groups, Sib. Mat. J., 3 (1962), 446–470 (in Russian).
[19] Shevrin, L.N.,Semigroups with Dedekind subsemi-group lattices, Dokl. Acad. Nauk SSSR, 148 (1963), 292–295 (in Russian).
[20] Shevrin, L.N. and A.J. Ovsyannikov,Semigroups and their subsemigroup lattices, Semigroup Forum, 27 (1983), 1–154. · Zbl 0523.20037
[21] Varlet, J.,Congruences dans les demi-lattis, Bull. Soc. roy. sci. Liege 34 (1965), 231–240. · Zbl 0148.01201
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.