Congruence extensions for semilattices with distributivity

Hickman, Robert

doi:10.1007/bf02488029

Cited by 2 publications

(7 citation statements)

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“…THEOREM terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700025374 [9] Mildly distributive semilattices 295 PROPOSITION …”

Section: Construction and Comparisonsmentioning

confidence: 99%

“…The main purpose of Hickman [9] was to investigate the relationship between the lattice of join partial congruences of a weakly distributive semilattice S (denoted &" (5)) and the lattice of lattice congruences of f"(S) (denoted ). The principal result was PROPOSITION …”

Section: The Lattice Of Join Partial Congruence Relationsmentioning

confidence: 99%

“…We recall a few definitions and results from Hickman PROOF. The statement of (ii) is not mentioned explicitly in Hickman [9]. Rather it is shown that g: <tf{H) -tf(S) defined by g((a u ... y a n ) H a ) = (a,,... ,a n ) s a is a lattice isomorphism.…”

Section: Finitely Derivable Mildly Distributive Semilatticesmentioning

confidence: 99%

“…The main purpose of Hickman [9] was to investigate the relationship between the lattice of join partial congruences of a weakly distributive semilattice S (denoted &" (5) Let S be a semilattice, a, b E S with a < b. Then a is said to be strong below b if a V y exists for all y < b.…”

Section: The Lattice Of Join Partial Congruence Relationsmentioning

confidence: 99%

“…LEMMA 7.1 (Hickman [9,Lemma 2.3]). Let S be a weakly distributive semilattice, a,,... ,a n G 5 and suppose (a,,... ,a n ) has an upper bound.…”

Section: Congruence Disrributivirymentioning

confidence: 99%

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Mildly distributive semilattices

Hickman

1984

J Aust Math Soc A

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There is no single generalization of distributivity to semilattices. This paper investigates the class of mildly distributive semilattices, which lies between the two most commonly discussed classes in this area-weakly distributive semilattices and distributive semilattices. Particular attention is paid to describing and characterizing congruence distributive mildly distributive semilattices, in contrast to distributive semilattices, whose lattice of join partial congruences is badly behaved and which are difficult to describe.

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“…THEOREM terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700025374 [9] Mildly distributive semilattices 295 PROPOSITION …”

Section: Construction and Comparisonsmentioning

confidence: 99%

Section: The Lattice Of Join Partial Congruence Relationsmentioning

confidence: 99%

Section: Finitely Derivable Mildly Distributive Semilatticesmentioning

confidence: 99%

Section: The Lattice Of Join Partial Congruence Relationsmentioning

confidence: 99%

“…LEMMA 7.1 (Hickman [9,Lemma 2.3]). Let S be a weakly distributive semilattice, a,,... ,a n G 5 and suppose (a,,... ,a n ) has an upper bound.…”

Section: Congruence Disrributivirymentioning

confidence: 99%

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