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61
The Reflection Theorem
 Journal of Formalized Mathematics
, 1990
"... this paper (and in another Mizar articles) we work in TarskiGrothendieck (TG) theory (see [17]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [15] and some concepts of MK are modeled. The concepts are ..."
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Cited by 249 (51 self)
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this paper (and in another Mizar articles) we work in TarskiGrothendieck (TG) theory (see [17]) which ensures the existence of sets that have properties like universal class (i.e. this theory is stronger than MK). The sets are introduced in [15] and some concepts of MK are modeled. The concepts are: the class On of all ordinal numbers belonging to the universe, subclasses, transfinite sequences of nonempty elements of universe, etc. The reflection theorem states that if A ¸ is an increasing and continuous transfinite sequence of nonempty sets and class A =
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 93 (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
Homomorphisms of many sorted algebras
 Journal of Formalized Mathematics
, 1994
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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).
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).
The product and the determinant of matrices with entries in a field
, 2003
"... Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field. ..."
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Cited by 27 (0 self)
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Concerned with a generalization of concepts introduced in [14], i.e. there are introduced the sum and the product of matrices of any dimension of elements of any field.
Cartesian Products of Relations and Relational Structures
, 1996
"... In this paper the definitions of cartesian products of relations and relational structures are introduced. Facts about these notions are proved. This work is the continuation of formalization of [8]. ..."
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Cited by 20 (6 self)
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In this paper the definitions of cartesian products of relations and relational structures are introduced. Facts about these notions are proved. This work is the continuation of formalization of [8].
The Equational Characterization of Continuous Lattices
, 2003
"... The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of [18] we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary i ..."
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Cited by 7 (0 self)
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The class of continuous lattices can be characterized by infinitary equations. Therefore, it is closed under the formation of subalgebras and homomorphic images. Following the terminology of [18] we introduce a continuous lattice subframe to be a sublattice closed under the formation of arbitrary infs and directed sups. This notion corresponds with a subalgebra of a continuous lattice in [16]. The class of completely distributive lattices is also introduced in the paper. Such lattices are complete and satisfy the most restrictive type of the general distributivity law. Obviously each completely distributive lattice is a Heyting algebra. It was hard to find the best Mizar implementation of the complete distributivity equational condition (denoted by CD in [16]). The powerful and well developed Many Sorted Theory gives the most convenient way of this formalization. A set double indexed by K, introduced in the paper, corresponds with a family {x j,k: j ∈ J,k ∈ K ( j)}. It is defined to be a suitable many sorted function. Two special functors: Sups and Infs as counterparts of Sup and Inf respectively, introduced in [33], are also defined. Originally the equation in Definition 2.4 of [16, p. 58] looks as follows: j∈J k∈K ( j)x j,k = � � f ∈M j∈Jx j, f ( j), where M is the set of functions defined on J with values f(j) ∈ K ( j).