Meet-irreducible elements in implicative lattices

Irreducible elements in multi-adjoint concept lattices

One of the most important elements in a lattice are the irreducible elements. For example, when the lattice is finite, which is usual in the computational case, it forms a base from which the complete lattice is obtained. These elements are also important in Formal Concept Analysis, since they are the basic information of a relational system. In the general fuzzy framework of multi-adjoint conc...

Meet – Continuous Lattices 1

The aim of this work is the formalization of Chapter 0 Section 4 of [11]. In this paper the definition of meet-continuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved. Let X, Y be non empty sets, let f be a function from X into Y , and let Z be a non empty subset of X. One can verify that f • Z is non empty. Let us note that every non empty relational structure which is reflexi...

Completely-Irreducible Elements

The following propositions are true: (1) For every sup-semilattice L and for all elements x, y of L holds ⌈⌉L(↑x∩ ↑y) = x ⊔ y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = x ⊓ y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L) \ ↑y, then ↑x \ {x} = ↑x ∩ ↑y. (4) Let L be a non empty relational structu...

Intervals in Lattices of -Meet-Closed Subsets

We study abstract properties of intervals in the complete lattice of all meet-closed subsets ( -subsemilattices) of a -(meet-)semilattice S, where is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closed set generated by A and x leaves a closed set). Such closure s...

Rigidity of Commensurators and Irreducible Lattices