Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part II

Kira Adaricheva; J. B. Nation

doi:10.1142/S021819671250066X

Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part II

Kira Adaricheva, J. B. Nation

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

Part I proved that for every quasivariety of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of (the dual of the lattice of sub-quasivarieties of ) is isomorphic to Con(S, +, 0, ). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group of operators acting on S, and each operator in fixes both 0 and 1, then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0, ).

Original language	English
Article number	1250066
Journal	International Journal of Algebra and Computation
Volume	22
Issue number	7
DOIs	https://doi.org/10.1142/S021819671250066X
Publication status	Published - Nov 2012

Keywords

Quasivariety
congruence lattice
representation
semilattice

ASJC Scopus subject areas

Mathematics(all)

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10.1142/S021819671250066X

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title = "Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part II",

abstract = "Part I proved that for every quasivariety of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of (the dual of the lattice of sub-quasivarieties of ) is isomorphic to Con(S, +, 0, ). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group of operators acting on S, and each operator in fixes both 0 and 1, then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0, ).",

keywords = "Quasivariety, congruence lattice, representation, semilattice",

author = "Kira Adaricheva and Nation, {J. B.}",

note = "Funding Information: The authors were supported in part by a grant from the U.S. Civilian Research and Development Foundation. The first author was also supported in part by INTAS Grant N03-51-4110.",

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AU - Adaricheva, Kira

AU - Nation, J. B.

N1 - Funding Information: The authors were supported in part by a grant from the U.S. Civilian Research and Development Foundation. The first author was also supported in part by INTAS Grant N03-51-4110.

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N2 - Part I proved that for every quasivariety of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of (the dual of the lattice of sub-quasivarieties of ) is isomorphic to Con(S, +, 0, ). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group of operators acting on S, and each operator in fixes both 0 and 1, then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0, ).

AB - Part I proved that for every quasivariety of structures (which may have both operations and relations) there is a semilattice S with operators such that the lattice of quasi-equational theories of (the dual of the lattice of sub-quasivarieties of ) is isomorphic to Con(S, +, 0, ). It is known that if S is a join semilattice with 0 (and no operators), then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0). We prove that if S is a semilattice having both 0 and 1 with a group of operators acting on S, and each operator in fixes both 0 and 1, then there is a quasivariety such that the lattice of theories of is isomorphic to Con(S, +, 0, ).

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