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The paper is the Mizar encoding of the chapter 0 section 3 of [12] In the paper the following concept are defined: Galois connections, Heyting algebras, and Boolean algebras.

The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of [14].

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[23], [6], [11], [7], and [15] provide the notation and terminology for this paper. The following proposition is true 1. PRELIMINARIES (1) For every non empty poset L and for every element x of L holds compactbelow(x) = ↓x∩the carrier of CompactSublatt(L). Let L be a non empty reflexive transitive relational structure and let X be a subset of 〈Ids(L),⊆〉. Then � X is a subset of L. Next we state a number of propositions: (2) For every non empty relational structure L and for all subsets X, Y of L such that X ⊆ Y holds finsups(X) ⊆ finsups(Y). (3) Let L be a non empty transitive relational structure, S be a sups-inheriting non empty full relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then finsups(X) ⊆ finsups(Y). (4) Let L be a complete transitive antisymmetric non empty relational structure, S be a supsinheriting non empty full relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then finsups(X) = finsups(Y). (5) Let L be a complete sup-semilattice and S be a join-inheriting non empty full relational substructure of L. Suppose ⊥L ∈ the carrier of S. Let X be a subset of L and Y be a subset of S. If X = Y, then finsups(Y) ⊆ finsups(X). (6) For every lower-bounded sup-semilattice L and for every subset X of 〈Ids(L),⊆ 〉 holds supX = ↓finsups ( � X). (7) For every reflexive transitive relational structure L and for every subset X of L holds ↓↓X =

In the paper open and order generating subsets are defined. Irreducible and prime elements are also defined. The article includes definitions and facts presented in

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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. MML Identifier: YELLOW_4

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.