-completeness and fixpoint properties
The concept of a -ideal in -distributive posets is introduced. Several properties of -ideals in -distributive posets are established. Further, the interrelationships between -ideals and -ideals in -distributive posets are investigated. Moreover, a characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of non-dense ideals. It is shown that every -ideal of a -distributive meet semilattice is semiprime. Several counterexamples are discussed.
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 -filter of a poset is contained in a proper semiprime filter, then it is -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...
The concept of -ideals in posets is introduced. Several properties of -ideals in -distributive posets are studied. Characterization of prime ideals to be -ideals in -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal of a -distributive poset is non-dense, then is an -ideal. Moreover, it is shown that the set of all -ideals of a poset with forms a complete lattice. A result analogous to separation theorem for finite -distributive...
In this paper, we generalize the notion of supremum and infimum in a poset.