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43
The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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Cited by 731 (68 self)
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Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on [9] and [10].
Partially Ordered Sets
, 2000
"... this article we define the choice function of a nonempty set family that does not contain ; as introduced in [6, pages 8889]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure nonempty set and order of the set, cha ..."
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Cited by 155 (4 self)
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this article we define the choice function of a nonempty set family that does not contain ; as introduced in [6, pages 8889]. We define order of a set as a relation being reflexive, antisymmetric and transitive in the set, partially ordered set as structure nonempty set and order of the set, chains, lower and upper cone of a subset, initial segments of element and subset of partially ordered set. Some theorems that belong rather to [5] or [14] are proved. MML Identifier: ORDERS1.
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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Cited by 52 (0 self)
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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.
Zermelo Theorem and Axiom of Choice
 Journal of Formalized Mathematics
, 1989
"... Summary. The article is continuation of [2] and [1], and the goal of it is show that Zermelo theorem (every set has a relation which well orders it proposition (26)) and axiom of choice (for every nonempty family of nonempty and separate sets there is set which has exactly one common element with ..."
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Cited by 39 (21 self)
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Summary. The article is continuation of [2] and [1], and the goal of it is show that Zermelo theorem (every set has a relation which well orders it proposition (26)) and axiom of choice (for every nonempty family of nonempty and separate sets there is set which has exactly one common element with arbitrary family member proposition (27)) are true. It is result of the Tarski’s axiom A introduced in [5] and repeated in [6]. Inclusion as a settheoretical binary relation is introduced, the correspondence of well ordering relations to ordinal numbers is shown, and basic properties of equinumerosity are presented. Some facts are based on [4]. MML Identifier: WELLORD2. WWW:
Filters  part II. Quotient lattices modulo filters and direct product of two lattices
 Formalized Mathematics
, 1991
"... Summary. Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, etc.). Based on it, the quotient (also called ..."
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Cited by 21 (9 self)
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Summary. Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, etc.). Based on it, the quotient (also called factor) lattice modulo a filter (i.e. modulo the equivalence relation w.r.t the filter) is introduced. Similarly, some properties of the direct product of two binary (unary) operations are present and then the direct product of two lattices is introduced. Besides, the heredity of distributivity, modularity, completeness, etc., for the product of lattices is also shown. Finally, the concept of isomorphic lattices is introduced, and there is shown that every Boolean lattice B is isomorphic with the direct product of the factor lattice B/[a] and the lattice latt[a], where a is an element of B.
Relations of Tolerance
, 2003
"... Introduces notions of relations of tolerance, tolerance set and neighbourhood of an element. The basic properties of relations of tolerance are proved. ..."
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Cited by 19 (0 self)
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Introduces notions of relations of tolerance, tolerance set and neighbourhood of an element. The basic properties of relations of tolerance are proved.
Vertex Sequences Induced by Chains
, 1996
"... In the three preliminary sections to the article we define two operations on finite sequences which seem to be of general interest. The first is the cut operation that extracts a contiguous chunk of a finite sequence from a position to a position. The second operation is a glueing catenation that ..."
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Cited by 18 (7 self)
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In the three preliminary sections to the article we define two operations on finite sequences which seem to be of general interest. The first is the cut operation that extracts a contiguous chunk of a finite sequence from a position to a position. The second operation is a glueing catenation that given two finite sequences catenates them with removal of the first element of the second sequence. The main topic of the article is to define an operation which for a given chain in a graph returns the sequence of vertices through which the chain passes. We define the exact conditions when such an operation is uniquely definable. This is done with the help of the so called twovalued alternating finite sequences. We also prove theorems about the existence of simple chains which are subchains of a given chain. In order to do this we define the notion of a finite subsequence of a typed finite sequence.