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491
Functions from a set to a set
 Journal of Formalized Mathematics
, 1989
"... function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] ..."
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Cited by 1089 (23 self)
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function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] in the new terminology. Also some basic facts about functions of two variables are proved.
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.
The Modification of a Function by a Function and the Iteration of the Composition of a Function
, 1990
"... ..."
Manysorted sets
 Journal of Formalized Mathematics
, 1993
"... Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is def ..."
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Cited by 189 (23 self)
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Summary. The article deals with parameterized families of sets. When treated in a similar way as sets (due to systematic overloading notation used for sets) they are called many sorted sets. For instance, if x and X are two manysorted sets (with the same set of indices I) then relation x ∈ X is defined as ∀i∈Ixi ∈ Xi. I was prompted by a remark in a paper by Tarlecki and Wirsing: “Throughout the paper we deal with manysorted sets, functions, relations etc.... We feel free to use any standard settheoretic notation without explicit use of indices ” [6, p. 97]. The aim of this work was to check the feasibility of such approach in Mizar. It works. Let us observe some peculiarities: empty set (i.e. the many sorted set with empty set of indices) belongs to itself (theorem 133), we get two different inclusions X ⊆ Y iff ∀i∈IXi ⊆ Yi and X ⊑ Y iff ∀xx ∈ X ⇒ x ∈ Y equivalent only for sets that yield non empty values. Therefore the care is advised.
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.
The sum and product of finite sequences of real numbers.
 Formalized Mathematics,
, 1990
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Many sorted algebras
 Journal of Formalized Mathematics
, 1994
"... Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between ..."
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Cited by 123 (14 self)
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Summary. The basic purpose of the paper is to prepare preliminaries of the theory of many sorted algebras. The concept of the signature of a many sorted algebra is introduced as well as the concept of many sorted algebra itself. Some auxiliary related notions are defined. The correspondence between (1 sorted) universal algebras [8] and many sorted algebras with one sort only is described by introducing two functors mapping one into the other. The construction is done this way that the composition of both functors is the identity on universal algebras.
Complete lattices
 Journal of Formalized Mathematics
, 1992
"... Summary. In the first section the lattice of subsets of distinct set is introduced. The join and meet operations are, respectively, union and intersection of sets, and the ordering relation is inclusion. It is shown that this lattice is Boolean, i.e. distributive and complementary. The second sectio ..."
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Cited by 119 (33 self)
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Summary. In the first section the lattice of subsets of distinct set is introduced. The join and meet operations are, respectively, union and intersection of sets, and the ordering relation is inclusion. It is shown that this lattice is Boolean, i.e. distributive and complementary. The second section introduces the poset generated in a distinct lattice by its ordering relation. Besides, it is proved that posets which have l.u.b.’s and g.l.b.’s for every two elements generate lattices with the same ordering relations. In the last section the concept of complete lattice is introduced and discussed. Finally, the fact that the function f from subsets of distinct set yielding elements of this set is a infinite union of some complete lattice, if f yields an element a for singleton {a} and f ( f ◦X) = f ( ⊔ X) for every subset X, is proved. Some concepts and proofs are based on [8] and [9].
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.