On the Group of Automorphisms of Universal Algebra and Many Sorted Algebra.pdf

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On the Group of Automorphisms of Universal Algebra and Many Sorted Algebra.pdf

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On the Group of Automorphisms of Universal Algebra and Many Sorted Algebra

JOURNAL OF FORMALIZED MATHEMATICSVolume 6, Released 1994, Published 2002Inst. of Computer Science, Univ. of Bia lystok On the Group of Automorphisms of UniversalAlgebra andMany Sorted AlgebraArtur Korni lowiczInstitute of MathematicsWarsaw UniversityBia lystokSummary. The aim of the article is to check the compatibility of the au-tomorphisms of universal algebras introduced in [9] and the corresponding conceptfor many sorted algebras introduced in [10].MML Identier: AUTALG_1.WWW: /JFM/Vol6/autalg_1.htmlThe articles [14], [21], [7], [20], [22], [5], [6], [2], [4], [15], [16], [12], [18], [19], [1], [11], [3],[9], [13], [17], [10], and [8] provide the notation and terminology for this paper.1. On the Group of Automorphisms of Universal AlgebraIn this paper U1 is a universal algebra and f , g are functions from U1 into U1.One can prove the following proposition(1) idthe carrier of U1 is an isomorphism of U1 and U1.Let us consider U1. The functor UAAut(U1) yields a non empty set of functions fromthe carrier of U1 to the carrier of U1 and is dened by the conditions (Def. 1).(Def. 1)(i) Every element of UAAut(U1) is a function from U1 into U1, and(ii) for every function h from U1 into U1 holds h 2 UAAut(U1) i h is an isomorphismof U1 and U1.One can prove the following propositions:(2) UAAut(U1) (the carrier of U1)the carrier of U1 :(4)1 idthe carrier of U1 2 UAAut(U1):(5) For all f , g such that f is an element of UAAut(U1) and g = f1 holds g is anisomorphism of U1 and U1.(6) For every element f of UAAut(U1) holds f1 2 UAAut(U1):1 The proposition (3) has been removed. 1 c Association of Mizar Users on the group of automorphisms of universal . . . 2(7) For all elements f1, f2 of UAAut(U1) holds f1 f2 2 UAAut(U1):Let us consider U1. The functor UAAutComp(U1) yields a binary operation onUAAut(U1) and is dened by:(Def. 2) For all elements x, y of UAAut(U1) holds (UAAutComp(U1))(x; y) = y x:Let us consider U1. The functor UAAutGroup(U1) yielding a group is dened