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Combining of Circuits
, 2002
"... this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S ..."
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Cited by 90 (24 self)
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this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S
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 86 (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 ∗.
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 63 (20 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).
Curried and uncurried functions
 Journal of Formalized Mathematics
, 1990
"... Summary. In the article following functors are introduced: the projections of subsets of the Cartesian product, the functor which for every function f: X ×Y → Z gives some curried function (X → (Y → Z)), and the functor which from curried functions makes uncurried functions. Some of their properties ..."
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Cited by 62 (19 self)
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Summary. In the article following functors are introduced: the projections of subsets of the Cartesian product, the functor which for every function f: X ×Y → Z gives some curried function (X → (Y → Z)), and the functor which from curried functions makes uncurried functions. Some of their properties and some properties of the set of all functions from a set into a set are also shown.
On the Decomposition of the States of SCM
, 1993
"... This article continues the development of the basic terminology ..."
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Cited by 52 (1 self)
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This article continues the development of the basic terminology
Binary operations on finite sequences
 Journal of Formalized Mathematics
, 1990
"... Summary. We generalize the semigroup operation on finite sequences introduced in [8] for binary operations that have a unity or for nonempty finite sequences. ..."
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Cited by 37 (2 self)
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Summary. We generalize the semigroup operation on finite sequences introduced in [8] for binary operations that have a unity or for nonempty finite sequences.
Subcategories and products of categories
 Journal of Formalized Mathematics
, 1990
"... inclusion functor is the injection (inclusion) map E ֒ → which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that HomE(a,b) = HomC(b,b) for any objects a,b of ..."
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Cited by 29 (1 self)
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inclusion functor is the injection (inclusion) map E ֒ → which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that HomE(a,b) = HomC(b,b) for any objects a,b of E, are defined. A subcategory E of C is full when the inclusion functor E ֒ → is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories B and C is constructed in the usual way. Moreover, some simple facts on bi f unctors (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors (complex functors and product functors).
Continuity of mappings over the union of subspaces
 Journal of Formalized Mathematics
, 1992
"... Summary. Let X and Y be topological spaces and let X1 and X2 be subspaces of X. Let f: X1 ∪ X2 → Y be a mapping defined on the union of X1 and X2 such that the restriction mappings f X1 and f X2 are continuous. It is well known that if X1 and X2 are both open (closed) subspaces of X, then f is con ..."
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Cited by 20 (4 self)
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Summary. Let X and Y be topological spaces and let X1 and X2 be subspaces of X. Let f: X1 ∪ X2 → Y be a mapping defined on the union of X1 and X2 such that the restriction mappings f X1 and f X2 are continuous. It is well known that if X1 and X2 are both open (closed) subspaces of X, then f is continuous (see e.g. [7, p.106]). The aim is to show, using Mizar System, the following theorem (see Section 5): If X1 and X2 are weakly separated, then f is continuous (compare also [14, p.358] for related results). This theorem generalizes the preceding one because if X1 and X2 are both open (closed), then these subspaces are weakly separated (see [6]). However, the following problem remains open. Problem 1. Characterize the class of pairs of subspaces X1 and X2 of a topological space X such that (∗) for any topological space Y and for any mapping f: X1 ∪ X2 → Y, f is continuous if the restrictions f X1 and f X2 are continuous. In some special case we have the following characterization: X1 and X2 are separated iff X1 misses X2 and the condition (∗) is fulfilled. In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer. Problem 2. Suppose the condition (∗) is fulfilled. Must X1 and X2 be weakly separated Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology.