On Uniquely Complemented Posets

Waphare, B. N.; Joshi, Vinayak

doi:10.1007/s11083-005-9002-0

Cited by 12 publications

(16 citation statements)

References 14 publications

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“…A detailed study of Boolean posets can be found also in Niederle [26] and Waphare and Joshi [37]. Further, it can be easily verified that a Boolean poset is pseudocomplemented and complemented.…”

Section: Poset Of Prime Idealsmentioning

confidence: 91%

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: Poset Of Prime Idealsmentioning

confidence: 91%

Prime ideals in 0-distributive posets

Joshi

Mundlik²

2013

Open Mathematics

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“…Uniquely complemented posets were treated by B. N. Waphare and V. V. Joshi ( [15]). We will use different tools and methods and state other conditions formulated in a different language.…”

Section: Proposition 24 ([11]) Every Uniquely Complemented Pseudocommentioning

confidence: 99%

“…An important progress in this direction was already obtained by B. N. Waphare and V. V. Joshi in [15]. They formulated such conditions by means of pseudocomplements and so-called regular elements.…”

Section: Introductionmentioning

confidence: 99%

Uniquely Complemented Posets

Chajda

Länger²,

Paseka

2017

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We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.

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“…Denote A = {a : a ∈ A}. We say that P satisfies the De Morgan laws if {{x, y} u } = {x , y } and {{x, y} } = {x , y } u for all x, y ∈ P ; see [3,23].…”

Section: Zero Divisor Graphs Of Boolean Posetsmentioning

confidence: 99%