Definitions and Properties of the Join and Meet of Subsets1

This paper is the continuation of formalization of [4]. The definitions of meet and join of subsets of relational structures are introduced. The properties of these notions are proved. [6] provide the notation and terminology for this paper. 1. PRELIMINARIES One can prove the following propositions: (1) Let L be a relational structure, X be a set, and a be an element of L. If a ∈ X and sup X exists in L, then a ≤ L X. (2) Let L be a relational structure, X be a set, and a be an element of L. If a ∈ X and inf X exists in L, then − L X ≤ a. Let L be a relational structure and let A, B be subsets of L. We say that A is finer than B if and only if: (Def. 1) For every element a of L such that a ∈ A there exists an element b of L such that b ∈ B and a ≤ b. We say that B is coarser than A if and only if: (Def. 2) For every element b of L such that b ∈ B there exists an element a of L such that a ∈ A and a ≤ b. Let L be a non empty reflexive relational structure and let A, B be subsets of L. Let us note that the predicate A is finer than B is reflexive. Let us note that the predicate B is coarser than A is reflexive. The following propositions are true: (3) For every relational structure L and for every subset B of L holds / 0 L is finer than B.