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Many sorted quotient algebra
 Journal of Formalized Mathematics
, 1994
"... Summary. This article introduces the construction of a many sorted quotient algebra. A few preliminary notions such as a many sorted relation, a many sorted equivalence relation, a many sorted congruence and the set of all classes of a many sorted relation are also formulated. ..."
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Summary. This article introduces the construction of a many sorted quotient algebra. A few preliminary notions such as a many sorted relation, a many sorted equivalence relation, a many sorted congruence and the set of all classes of a many sorted relation are also formulated.
Terms over Many Sorted Universal Algebra
 Journal of Formalized Mathematics
, 1994
"... Pure terms (without constants)... ..."
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On the Group of Automorphisms of Universal Algebra and Many Sorted Algebra
, 2002
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Extensions of Mappings on Generator Set
, 2001
"... this paper. 1. Preliminaries For simplicity, we adopt the following rules: S is a non void non empty many sorted signature, U 1 , U 2 , U 3 are nonempty algebras over S, I is a set, A is a many sorted set indexed by I , and B, C are nonempty many sorted sets indexed by I . The following four pr ..."
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this paper. 1. Preliminaries For simplicity, we adopt the following rules: S is a non void non empty many sorted signature, U 1 , U 2 , U 3 are nonempty algebras over S, I is a set, A is a many sorted set indexed by I , and B, C are nonempty many sorted sets indexed by I . The following four propositions are true: (1) For every binary relation R and for all sets X , Y such that X ` Y holds (R#Y ) ffi X = R ffi X: (2) Let A be a set, B, C be non empty sets, f be a function from A into B, g b
Certain Facts about Families of Subsets of Many Sorted Sets
, 2002
"... this paper. 1. Preliminaries For simplicity, we follow the rules: I, G, H, i are sets, A, B, M are many sorted sets indexed by I, s 1 , s 2 , s 3 are families of subsets of I, v, w are subsets of I, and F is a many sorted function indexed by I ..."
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this paper. 1. Preliminaries For simplicity, we follow the rules: I, G, H, i are sets, A, B, M are many sorted sets indexed by I, s 1 , s 2 , s 3 are families of subsets of I, v, w are subsets of I, and F is a many sorted function indexed by I
Examples of category structures
 Journal of Formalized Mathematics
, 1996
"... [10], and [11] provide the notation and terminology for this paper. The following proposition is true 1. PRELIMINARIES (1) For all sets X1, X2 and for all sets a1, a2 holds [:X1 ↦− → a1, X2 ↦− → a2:] = [:X1, X2:] ↦− → 〈a1, a2〉. Let I be a set. One can check that 0I is function yielding. Next we sta ..."
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[10], and [11] provide the notation and terminology for this paper. The following proposition is true 1. PRELIMINARIES (1) For all sets X1, X2 and for all sets a1, a2 holds [:X1 ↦− → a1, X2 ↦− → a2:] = [:X1, X2:] ↦− → 〈a1, a2〉. Let I be a set. One can check that 0I is function yielding. Next we state two propositions: (2) For all functions f, g holds �(g · f) = g · � f. (3) For all functions f, g, h holds � ( f · [:g, h:]) = � f · [:h, g:]. Let f be a function yielding function. Observe that � f is function yielding. Next we state the proposition (4) Let I be a set and A, B, C be many sorted sets indexed by I. Suppose A is transformable to B. Let F be a many sorted function from A into B and G be a many sorted function from B into C. Then G ◦ F is a many sorted function from A into C. Let I be a set and let A be a many sorted set indexed by [:I, I:]. Then �A is a many sorted set