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113
Finite Sequences and Tuples of Elements of a Nonempty Sets
, 1990
"... this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D ..."
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Cited by 332 (7 self)
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this article is the definition of tuples. The element of a set of all sequences of the length n of D is called a tuple of a nonempty set D and it is denoted by element of D
Binary operations applied to functions
 Journal of Formalized Mathematics
, 1989
"... Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The ..."
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Cited by 299 (43 self)
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Summary. In the article we introduce functors yielding to a binary operation its composition with an arbitrary functions on its left side, its right side or both. We prove theorems describing the basic properties of these functors. We introduce also constant functions and converse of a function. The recent concept is defined for an arbitrary function, however is meaningful in the case of functions which range is a subset of a Cartesian product of two sets. Then the converse of a function has the same domain as the function itself and assigns to an element of the domain the mirror image of the ordered pair assigned by the function. In the case of functions defined on a nonempty set we redefine the above mentioned functors and prove simplified versions of theorems proved in the general case. We prove also theorems stating relationships between introduced concepts and such properties of binary operations as commutativity or associativity.
A Borsuk theorem on homotopy types
 Journal of Formalized Mathematics
, 1991
"... Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retrac ..."
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Cited by 108 (6 self)
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Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.
Families of subsets, subspaces and mappings in topological spaces
 Journal of Formalized Mathematics
, 1989
"... Summary. This article is a continuation of [11]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topolo ..."
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Cited by 74 (2 self)
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Summary. This article is a continuation of [11]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological space. Notion of a family of complements of sets and a closed (open) family have been also introduced. Next some theorems refer to subspaces in a topological space: some facts about types in a subspace, theorems about open and closed sets and families in a subspace. A notion of restriction of a family has been also introduced and basic properties of this notion have been proved. The last part of the article is about mappings. There are proved necessary and sufficient conditions for a mapping to be continuous. A notion of homeomorphism has been defined next. Theorems about homeomorphisms of topological spaces have been also proved.
Binary operations applied to finite sequences.
 Formalized Mathematics,
, 1990
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Cartesian product of functions
 Journal of Formalized Mathematics
, 1991
"... Summary. A supplement of [3] and [2], i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two ..."
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Cited by 64 (22 self)
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Summary. A supplement of [3] and [2], i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two facts are presented: quasidistributivity of the power of the set to other one w.r.t. the union (X � x f (x) ≈ ∏x X f (x) ) and quasidistributivity of the product w.r.t. the raising to the power (∏x f (x) X ≈ (∏x f (x)) X).
Construction of rings and left, right, and bimodules over a ring
 Journal of Formalized Mathematics
, 1990
"... Summary. Definitions of some classes of rings and left, right, and bimodules over a ring and some elementary theorems on rings and skew fields. ..."
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Cited by 63 (16 self)
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Summary. Definitions of some classes of rings and left, right, and bimodules over a ring and some elementary theorems on rings and skew fields.
Families of Subsets, Subspaces and Mappings in Topological Spaces
"... This article is a continuation of [13]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological spa ..."
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Cited by 62 (0 self)
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This article is a continuation of [13]. Some basic theorems about families of sets in a topological space have been proved. Following redefinitions have been made: singleton of a set as a family in the topological space and results of boolean operations on families as a family of the topological space. Notion of a family of complements of sets and a closed (open) family have been also introduced. Next some theorems refer to subspaces in a topological space: some facts about types in a subspace, theorems about open and closed sets and families in a subspace. A notion of restriction of a family has been also introduced and basic properties of this notion have been proved. The last part of the article is about mappings. There are proved necessary and sufficient conditions for a mapping to be continuous. A notion of homeomorphism has been defined next. Theorems about homeomorphisms of topological spaces have been also proved. MML Identifier: TOPS2.
Joining of decorated trees
 Journal of Formalized Mathematics
, 1993
"... Summary. This is the continuation of the sequence of articles on trees (see [2], [4], [5]). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in [5]. We will also introduce the operation of substitution. In the last section we dealt with tre ..."
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Cited by 56 (19 self)
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Summary. This is the continuation of the sequence of articles on trees (see [2], [4], [5]). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in [5]. We will also introduce the operation of substitution. In the last section we dealt with trees decorated by Cartesian product, i.e. we showed some lemmas on joining operations applied to such trees.
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.