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A classical first order language
 Journal of Formalized Mathematics
, 1990
"... this paper. In this paper i, j, k are natural numbers. Let x, y, a, b be sets. The functor (x = y a,b) yields a set and is defined as follows: (Def. 1) (x = y a,b) = a, if x = y, b, otherwise. Let D be a non empty set, let x, y be sets, and let a, b be elements of D. Then (x = y a,b) is an e ..."
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Cited by 153 (0 self)
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this paper. In this paper i, j, k are natural numbers. Let x, y, a, b be sets. The functor (x = y a,b) yields a set and is defined as follows: (Def. 1) (x = y a,b) = a, if x = y, b, otherwise. Let D be a non empty set, let x, y be sets, and let a, b be elements of D. Then (x = y a,b) is an element of D
Topological Properties of Subsets in Real Numbers
 JOURNAL OF FORMALIZED MATHEMATICS
, 2002
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Function domains and Frænkel operator
 Journal of Formalized Mathematics
, 1990
"... Summary. We deal with a non–empty set of functions and a non–empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B A is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some th ..."
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Cited by 147 (18 self)
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Summary. We deal with a non–empty set of functions and a non–empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B A is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some theorems concerning these concepts are proved, as well as a number of schemes dealing with infinity and the Axiom of Choice. The article contains a number of schemes allowing for simple logical transformations related to terms constructed with the Frænkel Operator.
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].
The complex numbers
 Journal of Formalized Mathematics
, 1990
"... Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for ..."
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Cited by 118 (1 self)
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Summary. We define the set C of complex numbers as the set of all ordered pairs z = 〈a,b 〉 where a and b are real numbers and where addition and multiplication are defined. We define the real and imaginary parts of z and denote this by a = ℜ(z), b = ℑ(z). These definitions satisfy all the axioms for a field. 0C = 0 + 0i and 1C = 1 + 0i are identities for addition and multiplication respectively, and there are multiplicative inverses for each non zero element in C. The difference and division of complex numbers are also defined. We do not interpret the set of all real numbers R as a subset of C. From here on we do not abandon the ordered pair notation for complex numbers. For example: i 2 = (0+1i) 2 = −1+0i � = −1. We conclude this article by introducing two operations on C which are not field operations. We define the absolute value of z denoted by z  and the conjugate of z denoted by z ∗.
Basic functions and operations on functions
 Journal of Formalized Mathematics
, 1989
"... Summary. We define the following mappings: the characteristic function of a subset of a set, the inclusion function (injection or embedding), the projections from a Cartesian product onto its arguments and diagonal function (inclusion of a set into its Cartesian square). Some operations on functions ..."
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Cited by 114 (4 self)
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Summary. We define the following mappings: the characteristic function of a subset of a set, the inclusion function (injection or embedding), the projections from a Cartesian product onto its arguments and diagonal function (inclusion of a set into its Cartesian square). Some operations on functions are also defined: the products of two functions (the complex function and the more general productfunction), the function induced on power sets by the image and inverseimage. Some simple propositions related to the introduced notions are proved.
Directed sets, nets, ideals, filters, and maps
 Journal of Formalized Mathematics
, 1996
"... Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8 ..."
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Cited by 109 (29 self)
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Summary. Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8, p. 2]), the definition of ideals and filters and the definition of maps preserving arbitrary and directed sups and arbitrary and filtered infs (1.9 also in [8, p. 4]). The concepts of semilattices, supsemiletices and poset lattices (1.8 in [8, p. 4]) are also introduced. A number of facts concerning the above notion and including remarks 1.4, 1.5, and 1.10 from [8, pp. 3–5] is presented.