Abstract
The duality between congruence lattices of semilattices, and algebraic subsets of an algebraic lattice, is extended to include semilattices with operators. For a set G of operators on a semilattice S, we have \({{\rm Con}(S,+,0,G) \cong^{d} {{\rm S}_{p}}(L,H)}\), where L is the ideal lattice of S, and H is a corresponding set of adjoint maps on L. This duality is used to find some representations of lattices as congruence lattices of semilattices with operators. It is also shown that these congruence lattices satisfy the Jónsson–Kiefer property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adaricheva, K., and J. Nation, Lattices of quasi-equational theories as congruence lattices of semilattices with operators parts I and II, International Journal of Algebra and Computation 22:N7, 2012.
Adaricheva K., Maróti M., McKenzie R., Nation J. B., Zenk E.: The Jónsson–Kiefer property, Studia Logica 83, 111–131 (2006)
Davey B., Jackson M., Pitkethly J., Talukder M.: Natural dualities for semilattice-based algebras, Algebra Universalis 57, 463–490 (2007)
Fajtlowicz, S., and J. Schmidt, Bézout Families, Join Congruences and Meet-Irreducible Ideals, in Lattice Theory (Proc. Colloq., Szeged, 1974), Colloq. Math. Soc. János Bolyai 14, North Holland, Amsterdam, 1976, pp. 51–76.
Freese R., Nation J.: Congruence lattices of semilattices, Pacific Journal of Mathematics 49, 51–58 (1973)
Gorbunov V.: The structure of lattices of quasivarieties, Algebra Universalis 32, 493–530 (1994)
Gorbunov, V., Algebraic Theory of Quasivarieties, Siberian School of Algebra and Logic, Plenum, New York, 1998.
Gorbunov, V., and V. Tumanov, Construction of Lattices of Quasivarieties, Math. Logic and Theory of Algorithms, Trudy Inst. Math. Sibirsk. Otdel. Adad. Nauk SSSR 2, Nauka, Novosibirsk, 1982, pp. 12–44.
Hoehnke, H.-J., Fully Invariant Algebraic Closure Systems of Congruences and Quasivarieties of Algebras, Lectures in Universal Algebra (Szeged, 1983), Colloq. Math. Soc. János Bolyai 43, North-Holland, Amsterdam, 1986, pp. 189–207.
Jónsson B., Kiefer J. E.: Finite sublattices of a free lattice, Canadian Journal of Mathematics 14, 487–497 (1962)
Papert D.: Congruence relations in semilattices, Journal of the London Mathematical Society 39, 723–729 (1964)
Schmidt E. T.: Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied 18, 3–20 (1968)
Schmidt, E. T., Kongruenzrelationen Algebraischer Strukturen, Mathematische Forschungsberichte, XXV VEB Deutscher Verlag der Wissenschaften, Berlin, 1969.
Tumanov V.: Finite distributive lattices of quasivarieties, Algebra and Logic 22, 119–129 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hyndman, J., Nation, J.B. & Nishida, J. Congruence Lattices of Semilattices with Operators. Stud Logica 104, 305–316 (2016). https://doi.org/10.1007/s11225-015-9641-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-015-9641-0