On Prime Semilattices

Pawar, Y. S.; Thakare, N. K.

doi:10.4153/cmb-1980-040-1

Cited by 5 publications

(18 citation statements)

References 4 publications

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“…The equivalence between .1/ and .2/ was proved by G. Grätzer in [10]. The equivalence between the condition .1/ and .4/ of Theorem 2.2 was given by J. Varlet in [14] and [13]. This result provides a characterization of distributivity of a semilattice through a separation property and generalizes the Stone's theorem for distributive lattices.…”

Section: Preliminariesmentioning

confidence: 76%

“…Recall that a bounded distributive semilattice A is normal if each irreducible filter P is contained in a unique maximal filter [13]. It is clear that a bounded distributive lattice is normal iff it is normal as a bounded distributive semilattice (see [8]).…”

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

“…It is clear that a bounded distributive lattice is normal iff it is normal as a bounded distributive semilattice (see [8]). While studying normal bounded distributive semilattices Pawar and Lokhande [13] have presented several characterizations of normal bounded distributive semilattices. The following result is proved in [13], but we give here a different proof for completeness.…”

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

“…While studying normal bounded distributive semilattices Pawar and Lokhande [13] have presented several characterizations of normal bounded distributive semilattices. The following result is proved in [13], but we give here a different proof for completeness. This result we shall need in Proposition 4.7.…”

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

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Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

Celani

2014

Open Mathematics

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Abstract:In this paper we shall study a notion of relative annihilator-preserving congruence relation and relative annihilator-preserving homomorphism in the class of bounded distributive semilattices. We shall give a topological characterization of this class of semilattice homomorphisms. We shall prove that the semilattice congruences that are associated with filters are exactly the relative annihilator-preserving congruence relations.

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Section: Preliminariesmentioning

confidence: 76%

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

Section: Relative Annihilators In Ds 01mentioning

confidence: 99%

See 2 more Smart Citations

Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

Celani

2014

Open Mathematics

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“…We remark that the concepts of prime distributive lattices and very weakly distributive lattices have been introduced and discussed in [7] and [8]. It is natural to ask whether the prime pseudo-complemented distributivity and the weakly pseudocomplemented distributivity are equivalent condition for a semilattice or a lattice?…”

Section: Resultsmentioning

confidence: 99%

Boolean Filters of Distributive Lattices

Rao

Shum

2013

International Journal of Mathematics and Soft Computing

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In this paper we introduce the notion of Boolean filters in a pseudo-complemented distributive lattice and characterize the class of all Boolean filters. Further a set of equivalent conditions are derived for a proper filter to become a prime Boolean filter. Also a set of equivalent conditions is derived for a pseudo-complemented distributive lattice to become a Boolean algebra. Finally, a Boolean filter is characterized in terms of congruences.

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Representable posets

Egrot

2016

Journal of Applied Logic

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A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals α and β a poset is said to be (α, β)-representable if an embedding into a field of sets exists that preserves meets of sets smaller than α and joins of sets smaller than β. We show using an ultraproduct/ultraroot argument that when 2 ≤ α, β ≤ ω the class of (α, β)-representable posets is elementary, but does not have a finite axiomatization in the case where either α or β = ω. We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.

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On Prime Semilattices

Cited by 5 publications

References 4 publications

Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

Boolean Filters of Distributive Lattices

Representable posets

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