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0 -ideals in 0 -distributive posets

Khalid A. Mokbel (2016)

Mathematica Bohemica

The concept of a 0 -ideal in 0 -distributive posets is introduced. Several properties of 0 -ideals in 0 -distributive posets are established. Further, the interrelationships between 0 -ideals and α -ideals in 0 -distributive posets are investigated. Moreover, a characterization of prime ideals to be 0 -ideals in 0 -distributive posets is obtained in terms of non-dense ideals. It is shown that every 0 -ideal of a 0 -distributive meet semilattice is semiprime. Several counterexamples are discussed.

0-distributive posets

Khalid A. Mokbel, Vilas S. Kharat (2013)

Mathematica Bohemica

Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper l -filter of a poset is contained in a proper semiprime filter, then it is 0 -distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that...

α -filters and α -order-ideals in distributive quasicomplemented semilattices

Ismael Calomino, Sergio A. Celani (2021)

Commentationes Mathematicae Universitatis Carolinae

We introduce some particular classes of filters and order-ideals in distributive semilattices, called α -filters and α -order-ideals, respectively. In particular, we study α -filters and α -order-ideals in distributive quasicomplemented semilattices. We also characterize the filters-congruence-cokernels in distributive quasicomplemented semilattices through α -order-ideals.

α -ideals in 0 -distributive posets

Khalid A. Mokbel (2015)

Mathematica Bohemica

The concept of α -ideals in posets is introduced. Several properties of α -ideals in 0 -distributive posets are studied. Characterization of prime ideals to be α -ideals in 0 -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal I of a 0 -distributive poset is non-dense, then I is an α -ideal. Moreover, it is shown that the set of all α -ideals α Id ( P ) of a poset P with 0 forms a complete lattice. A result analogous to separation theorem for finite 0 -distributive...

λ -lattices

Václav Snášel (1997)

Mathematica Bohemica

In this paper, we generalize the notion of supremum and infimum in a poset.

σ-Entangled linear orders and narrowness of products of Boolean algebras

Saharon Shelah (1997)

Fundamenta Mathematicae

We investigate σ-entangled linear orders and narrowness of Boolean algebras. We show existence of σ-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.

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