Ideals in atomic posets

Erné, Marcel; Joshi, Vinayak

doi:10.1016/j.disc.2015.01.001

Cited by 8 publications

(4 citation statements)

References 35 publications

(80 reference statements)

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“…Next we also need to quote the following useful terminologies. We refer the reader to [8,9]. Let (Q, ≤) and (Q , ≤) be two partial ordered sets (posets).…”

Section: Preliminariesmentioning

confidence: 99%

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

ZAMAN

2024

Hacettepe Journal of Mathematics and Statistics

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This article encompasses the study of non-regular semigroups. First, we show that the $\lambda$-semidirect product $M \rtimes^{\lambda} W$ of a left $P$-Ehresmann semigroup $M$ and a left restriction semigroup $W$ is a left $P$-Ehresmann semigroup. We explore the behavior of generalized Green's relations on $M \rtimes^{\lambda} W$, and investigate some properties of $M \rtimes^{\lambda} W$. Second, the Zappa-Szép product of a right Ehresmann semigroup and its distinguished semilattice is studied. Lastly, the theory of representations of left Ehresmann semigroups with zero via homomorphisms of left Ehresmann semigroups with zero into Clifford restriction semigroups with zero is presented.

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“…Next we also need to quote the following useful terminologies. We refer the reader to [8,9]. Let (Q, ≤) and (Q , ≤) be two partial ordered sets (posets).…”

Section: Preliminariesmentioning

confidence: 99%

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

ZAMAN

2024

Hacettepe Journal of Mathematics and Statistics

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“…These specific notions of polars fit into the general framework of polarities proposed by Birkhoff [5] (cf. [16,17,18]).…”

Section: Bmentioning

confidence: 99%

Tensor products and relation quantales

Erné¹,

Picado

2017

Algebra Univers.

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A classical tensor product A ⊗ B of complete lattices A and B, consisting of all down-sets in A × B that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A, B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.

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“…During last few years, the theory of prime ideals of posets has been developed; see David and Erné (1992), Erné (2006), Erné and Joshi (2015), Halaš et al (2010), Halaš and Rachůnek (1995) and, Joshi and Mundlik (2013). In fact, Venkatanarasimhan (1970) seems to be first who introduced the topology on the set of all prime semi-ideals of a poset and characterized T 1 -spaces.…”

Section: Introductionmentioning

confidence: 97%

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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Ideals in atomic posets

Cited by 8 publications

References 35 publications

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

On non-regular semigroups: $\lambda$-semidirect products, Zappa-Szép products and representations

Tensor products and relation quantales

The hull-kernel topology on prime ideals in posets

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