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254
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 114 (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 ∗.
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 95 (25 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
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 66 (21 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).
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
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).
Products and Coproducts in Categories
, 1992
"... this paper. 1. Indexed families ..."
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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.