Perfect semilattices.
Permutability of congruences in varieties with idempotent operations
Permutability of distributive congruence relations in a joint semilattice directed below
Polars in autometrized algebras
Prime ideals in 0-distributive posets
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...
Prime ideals in the lattice of additive induced-hereditary graph properties
An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups,...
Prime ideals yield almost maximal ideals
Primeness and semiprimeness in posets
The concept of a semiprime ideal in a poset is introduced. Characterizations of semiprime ideals in a poset as well as characterizations of a semiprime ideal to be prime in are obtained in terms of meet-irreducible elements of the lattice of ideals of and in terms of maximality of ideals. Also, prime ideals in a poset are characterized.
Principal congruence relations and principal tolerances on varieties of lattices
Principal tolerances on lattices
Pseudocompactness and the cozero part of a frame
A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a -frame and to Alexandroff spaces.