Notes on mildly distributive semilattices

In this paper we shall investigate the mildly distributive meet-semilattices by means of the study of their filters and Frink-ideals as well as applying the theory of annihilator. We recall some characterizations of the condition of mildly-distributivity and we give several new characterizations. We prove that the definition of strong free distributive extension, introduced by Hickman in 1984, can be simplified and we show a correspondence between (prime) Frink-ideals of a mildly distributive semilattice and (prime) ideals of its strong free distributive extension.

Baer ideals in 0-distributive posets

In this paper, we study Baer ideals in posets and obtain some characterizations of Baer ideals in [Formula: see text]-distributive posets. Further, we prove that in an ideal-distributive poset, every ideal is Baer (normal) if and only if every prime ideal is Baer (normal). We extend the concept of a quasicomplement to posets and prove characterizations of quasicomplemented poset. This extend the results regarding Baer ideals and quasicomplemented lattices mentioned in [Y. S. Pawar and S. S. Khopade, [Formula: see text]-ideals and annihilator ideals in 0-distributive lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49(1) (2010) 63–74; Y. S. Pawar and D. N. Mane, [Formula: see text]-ideals in 0-distributive semilattices and 0-distributive lattices, Indian J. Pure Appl. Math. 24(7–8) (1993) 435–443] to posets.

Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

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.

CONGRUENCE LIFTING OF SEMILATTICE DIAGRAMS

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted Mn). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES

We prove that for every distributive 〈∨,0〉-semilattice S, there are a meet-semilattice P with zero and a map μ: P × P → S such that μ(x,z) ≤ μ(x,y) ∨ μ(y,z) and x ≤ y implies that μ(x,y) = 0, for all x, y, z ∈ P, together with the following conditions:. (P1) μ(v,u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if μ(v,u) ≤ a ∨ b, then there are a positive integer n and a decomposition u = x0 ≤ x1 ≤ ⋯ ≤ xn = v such that either μ(xi + 1, xi) ≤ a or μ(xi+1, xi) ≤ b, for each i < n. (P3) The subset {μ(x,0) | x ∈ P} generates the semilattice S. Furthermore, every finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive 〈∨,0,1〉-semilattices.