pm-lattices

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

doi:10.1007/bf02485435

Cited by 8 publications

(9 citation statements)

References 3 publications

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“…The last section extends the results of Pawar and Thakare (1977), Balasubramani (2008) for pm-(semi)lattices, that is, bounded distributive (semi)lattices in which every prime ideal is contained in a unique maximal ideal. In the last section, we extend the following Theorem 1.1 of Balasubramani (2008) to posets.…”

Section: Introductionmentioning

confidence: 86%

“…Hence from the proof of Theorem 4.19, we have the following corollary. Corollary 4.20 (Pawar and Thakare 1977) Let Q be a bounded distributive lattice. Then the following conditions are equivalent:…”

Section: Corollary 412 (Balasubramani 2008) If Q Is a Bounded Pseudomentioning

confidence: 98%

“…By the hypothesis, there exist a / ∈ M 1 and b / ∈ M 2 such that (a, b) ⊆ M∈Max(Q) M. Clearly, Max(Q)\U ((a]) and Max(Q)\U ((b]) are the neighborhoods of M 1 and M 2 respectively in Max(Q). As (a, b) ⊆ Pawar and Thakare (1977) introduced the concept of a pm-lattice, that is, a bounded distributive lattice in which every prime ideal is contained in a unique maximal ideal. In the following theorem, Pawar (1978) gave a characterization of pm-lattices.…”

Section: Corollary 412 (Balasubramani 2008) If Q Is a Bounded Pseudomentioning

confidence: 99%

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The hull-kernel topology on prime ideals in posets

Mundlik¹,

Joshi

Halaš

2016

Soft Comput

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In this paper, we continue our study of prime ideals in posets that was started in Joshi and Mundlik (Cent Eur J Math 11(5):940-955, 2013) and, Erné and Joshi (Discrete Math 338:954-971, 2015). We study the hull-kernel topology on the set of all prime ideals P(Q), minimal prime ideals Min(Q) and maximal ideals Max(Q) of a poset Q. Then topological properties like compactness, connectedness and separation axioms of P(Q) are studied. Further, we focus on the space of minimal prime ideals Min(Q) of a poset Q. Under the additional assumption that every maximal ideal is prime, the collection of all maximal ideals Max(Q) of a poset Q forms a subspace of P(Q). Finally, we prove a characterization of a space of maximal ideals of a poset to be a normal space.

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Section: Introductionmentioning

confidence: 86%

Section: Corollary 412 (Balasubramani 2008) If Q Is a Bounded Pseudomentioning

confidence: 98%

Section: Corollary 412 (Balasubramani 2008) If Q Is a Bounded Pseudomentioning

confidence: 99%

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The hull-kernel topology on prime ideals in posets

Mundlik¹,

Joshi

Halaš

2016

Soft Comput

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“…The concept of a pm-lattice is introduced by Pawar, and Thakare (1977) as a bounded distributive lattice in which each prime ideal is contained in a unique maximal prime ideal. In the following theorem a characterization of pm-lattices are given.…”

Section: A Proper Filtermentioning

confidence: 99%

The hull-kernel topology on prime filters in residuated lattices

Rasouli

Dehghani

2021

Soft Comput

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The notion of hull-kernel topology on a collection of prime filters in a residuated lattice is introduced and investigated. It is observed that any collection of prime filters is a T0 topological space under the hull-kernel and the dual hull-kernel topologies. It is proved that any collection of prime filters is a T1 space if and only if it is an antichain, and it is a Hausdorff space if and only if it satisfies a certain condition. Some characterizations in which maximal filters forms a Hausdorff space are given. At the end, it is focused on the space of minimal prim filters, and it is shown that this space is a totally disconnected Hausdorff space. This paper is closed by a discussion abut the various forms of compactness and connectedness of this space.

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“…Johnstone (1982); Contessa (1982Contessa ( , 1984; Al-Ezeh (1989, 1990a; Banaschewski (2000); Aghajani and Tarizadeh (2020). Motivated by De Marco and Orsatti (1971) the notion of a pm-lattice was introduced by Pawar and Thakare (1977) as a bounded distributive lattice in which any prime ideal is contained in a unique maximal ideal. Simmons (1980, p. 185) showed that a bounded distributive lattice is a pm-lattice if and only if it is a normal lattice.…”

Section: Introductionmentioning

confidence: 99%

Gelfand residuated lattices

Rasouli¹,

Dehghani²

2022

Preprint

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In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand's residuated lattices strongly tied up with the hull-kernel topology. Especially, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull-kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull-kernel and the D-topology coincide on the set of maximal filters. 1

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pm-lattices

Cited by 8 publications

References 3 publications

The hull-kernel topology on prime ideals in posets

The hull-kernel topology on prime ideals in posets

The hull-kernel topology on prime filters in residuated lattices

Gelfand residuated lattices

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