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84
Galois Connections
, 1997
"... 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. ..."
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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.
MooreSmith Convergence
, 2003
"... 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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Cited by 32 (2 self)
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The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of [14].
Bases of Continuous Lattices
"... [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 ..."
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Cited by 16 (2 self)
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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 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). (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 supsemilattice and S be a joininheriting 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 lowerbounded supsemilattice 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 =
Irreducible and Prime Elements
, 2003
"... 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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Cited by 16 (0 self)
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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
Definitions and Properties of the Join and Meet of Subsets
, 1996
"... 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 ..."
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Cited by 12 (3 self)
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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
Meet  Continuous Lattices
, 1997
"... The aim of this work is the formalization of Chapter 0 Section 4 of [11]. In this paper the definition of meetcontinuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved. ..."
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Cited by 12 (4 self)
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The aim of this work is the formalization of Chapter 0 Section 4 of [11]. In this paper the definition of meetcontinuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved.