Geodesic
Representation of distributive algebraic spatial lattices by congruence lattices of semigroups and groupoids
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 818-831 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that every distributive algebraic spatial lattice is isomorphic to the congruence lattice of some semigroup and also of some groupoid with zero satisfying identities $x^2=0$ and $xy=yx$.
Keywords: congruence lattice, semigroup, spatial lattice.
Mots-clés : groupoid 
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A. L. Popovich. Representation of distributive algebraic spatial lattices by congruence lattices of semigroups and groupoids. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 818-831. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a27/

[1] C. J. Ash, “The lattice of ideals of a semigroup”, Algebra Universalis, 10 (1980), 395–398 | DOI | MR | Zbl

[2] R. Freese, W. A. Lampe, W. Taylor, “Congruence lattices of algebras of fixed similarity type, I”, Pacific J. Math., 82:1 (1979), 59–68 | DOI | MR | Zbl

[3] G. Grätzer, Lattice Theory: Foundation, Birkhäuser Verlag, Basel, 2011 | MR | Zbl

[4] G. Grätzer, Universal Algebra, Second edition with updates, Springer science+Business Media, LLC, 2008 | MR | Zbl

[5] G. Grätzer, E. T. Schmidt, “Characterizations of congruence lattices of abstract algebras”, Acta Sci. Math. (Szeged), 24 (1963), 34–59 | MR

[6] R. Egrot, R. Hirsh, “Completely representable lattices”, Algebra Univers, 67:3 (2012), 205–217 | DOI | MR | Zbl

[7] J. Johnson, R. L. Seifer, A survey of multi-unary algebras, Mimeographed seminar notes, U.C. Berkeley, 1967, 26 pp.

[8] B. Jonsson, Topics in Universal Algebra, Lecture Notes in Mathematics, 250, Springer-Verlag, 1972 | DOI | MR | Zbl

[9] W. A. Lampe, “Congruence lattices of algebras of fixed similarity type, II”, Pacific J. Math., 103:2 (1982), 475–508 | DOI | MR | Zbl

[10] W. A. Lampe, “Results and problems on congruence lattice representations”, Algebra univers., 55 (2006), 127–135 | DOI | MR | Zbl

[11] V. Repnitskiǐ, J. Tůma, “Intervals in subgroup lattices of countable locally finite groups”, Algebra univers., 59 (2008), 49–71 | DOI | MR | Zbl

[12] P. Ružička, J. Tůma, F. Wehrung, “Distributive congruence lattices of congruence-permutable algebras”, Journal of Algebra, 311:1 (2007), 96–116 | DOI | MR | Zbl

[13] E. T. Schmidt, “The ideal lattice of a distributive lattice with $0$ is the congruence lattice of a lattice”, Acta Sci. Math. (Szeged), 43 (1981), 153–168 | MR | Zbl

[14] J. Tůma, “Semilattice-valued measures”, Contr. Gen. Alg., 18 (2007)

[15] P. Zhu, “On Rees congruence semigroups”, Northeast. Math. J., 8 (1992), 185–191 | MR | Zbl