We presents some relations between the (maximal) spectre of a residuated lattice and the residuated lattices of its regular elements. We note the characterization found for the radical of a residuated lattice via the radical of the residuated lattices of the ragular elements. Finally, this last result is applied in the study of the simplicity and semi-simplicity of a residuated lattice.
Algorithms for generating random posets, random lattices and random lattice terms are given.
In this paper we shall study a notion of relative annihilator-preserving congruence relation and relative annihilator-preserving homomorphism in the class of bounded distributive semilattices. We shall give a topological characterization of this class of semilattice homomorphisms. We shall prove that the semilattice congruences that are associated with filters are exactly the relative annihilator-preserving congruence relations.
The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. -radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of -radicals. In addition, a necessary and sufficient condition for the equality...
We give some necessary and sufficient conditions for the Scott topology on a complete lattice to be sober, and a sufficient condition for the weak topology on a poset to be sober. These generalize the corresponding results in [1], [2] and [4].