Standard completions for quasiordered sets

Erné, Marcel; Wilke, Gerhard

doi:10.1007/bf02572747

Cited by 24 publications

(21 citation statements)

References 15 publications

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“…But the completion by cuts of a finite bounded distributive poset is pseudocomplemented; see Joshi [18]. Erné and Wilke [10] proved that the completion by cuts of a pseudocomplemented poset is pseudocomplemented.…”

Section: Corollary 32mentioning

confidence: 99%

Prime ideals in 0-distributive posets

Joshi

Mundlik²

2013

Open Mathematics

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In the first section of this paper, we prove an analogue of Stone's Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals. MSC:06B10, 06A12

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Section: Corollary 32mentioning

confidence: 99%

Prime ideals in 0-distributive posets

Joshi

Mundlik²

2013

Open Mathematics

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“…A finite poset can be visualized through its Hasse diagram (discrete graphs), which depicts the ordering relation [35]. This area of order theory was investigated in a series of papers by Erné [16,18] and independently by Chajda, Halas, Larmerová, Rachånek, Niederle [8,[26][27][28][29]31], and later by Joshi, Kharat, Mokbel, Mundlik, Waphare [33,45,46] and many others. In [19], they are mainly interested in ideal-theoretic properties and various degrees of (finite or infinite) distributivity in atomic posets.…”

Section: Introductionmentioning

confidence: 99%

A new view of relationship between atomic posets and complete (algebraic) lattices

2017

Open Mathematics

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“…Down-set lattices play a role in diverse areas of order theory and topology, for example, in the theory of standard completions for ordered sets [3,17,27], in connection with continuous and algebraic lattices [35], and also in the more general context of Zcontinuous and Z-algebraic posets [10,20,24,25]. For further categorical aspects of the down-set functor, see [8].…”

Section: Introduction: the Role Of Down-set Lattices In Dualitiesmentioning

confidence: 99%

Complete Congruences on Topologies and Down-set Lattices

Erné

Gehrke

Pultr

2006

Appl Categor Struct

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From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence on the frame of open sets is induced by a unique subspace A so that = {(U, V) | U ∩ A = V ∩ A}, and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.

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Standard completions for quasiordered sets

Cited by 24 publications

References 15 publications

Prime ideals in 0-distributive posets

Prime ideals in 0-distributive posets

A new view of relationship between atomic posets and complete (algebraic) lattices

Complete Congruences on Topologies and Down-set Lattices

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