Volume 6, 1994

University of Bialystok

Copyright (c) 1994 Association of Mizar Users

### The abstract of the Mizar article:

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

**by****Artur Kornilowicz**- Received December 13, 1994
- MML identifier: AUTALG_1

- [ Mizar article, MML identifier index ]

environ vocabulary UNIALG_1, FUNCT_1, GROUP_6, ALG_1, RELAT_1, FRAENKEL, FUNCT_2, BINOP_1, REALSET1, VECTSP_1, GROUP_1, PBOOLE, NATTRA_1, BOOLE, FUNCOP_1, PRALG_1, MSUALG_3, ZF_REFLE, AMI_1, MSUALG_1, CARD_3, MSUHOM_1, CQC_SIM1, WELLORD1, AUTALG_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, STRUCT_0, FINSEQ_1, BINOP_1, FRAENKEL, PBOOLE, VECTSP_1, GROUP_1, GROUP_6, CARD_3, UNIALG_1, UNIALG_2, ALG_1, PRALG_1, MSUALG_1, MSUALG_3, MSUHOM_1; constructors BINOP_1, FRAENKEL, GROUP_6, ALG_1, MSUALG_3, MSUHOM_1, MEMBERED, XBOOLE_0; clusters FRAENKEL, FUNCT_1, GROUP_1, MSUALG_1, PBOOLE, PRALG_1, RELSET_1, STRUCT_0, SUBSET_1, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1, XBOOLE_0, NUMBERS, ORDINAL2; requirements BOOLE, SUBSET; begin :: On the Group of Automorphisms of Universal Algebra reserve UA for Universal_Algebra, f, g for Function of UA, UA; theorem :: AUTALG_1:1 id the carrier of UA is_isomorphism UA, UA; definition let UA; func UAAut UA -> FUNCTION_DOMAIN of the carrier of UA, the carrier of UA means :: AUTALG_1:def 1 (for f be Element of it holds f is Function of UA, UA) & for h be Function of UA, UA holds h in it iff h is_isomorphism UA, UA; end; theorem :: AUTALG_1:2 UAAut UA c= Funcs (the carrier of UA, the carrier of UA); canceled; theorem :: AUTALG_1:4 id the carrier of UA in UAAut UA; theorem :: AUTALG_1:5 for f, g st f is Element of UAAut UA & g = f" holds g is_isomorphism UA, UA; theorem :: AUTALG_1:6 for f be Element of UAAut UA holds f" in UAAut UA; theorem :: AUTALG_1:7 for f1, f2 be Element of UAAut UA holds f1 * f2 in UAAut UA; definition let UA; func UAAutComp UA -> BinOp of UAAut UA means :: AUTALG_1:def 2 for x, y be Element of UAAut UA holds it.(x, y) = y * x; end; definition let UA; func UAAutGroup UA -> Group equals :: AUTALG_1:def 3 HGrStr (# UAAut UA, UAAutComp UA #); end; definition let UA; cluster UAAutGroup UA -> strict; end; theorem :: AUTALG_1:8 for x, y be Element of UAAutGroup UA for f, g be Element of UAAut UA st x = f & y = g holds x * y = g * f; theorem :: AUTALG_1:9 id the carrier of UA = 1.UAAutGroup UA; theorem :: AUTALG_1:10 for f be Element of UAAut UA for g be Element of UAAutGroup UA st f = g holds f" = g"; begin :: Some properties of Many Sorted Functions reserve I for set, A, B, C for ManySortedSet of I; definition let I, A, B; pred A is_transformable_to B means :: AUTALG_1:def 4 for i be set st i in I holds B.i = {} implies A.i = {}; reflexivity; end; theorem :: AUTALG_1:11 A is_transformable_to B & B is_transformable_to C implies A is_transformable_to C; theorem :: AUTALG_1:12 for x be set, A be ManySortedSet of {x} holds A = {x} --> A.x; theorem :: AUTALG_1:13 for F, G, H be Function-yielding Function holds (H ** G) ** F = H ** (G ** F); theorem :: AUTALG_1:14 for A, B be non-empty ManySortedSet of I for F be ManySortedFunction of A, B st F is "1-1" "onto" holds F"" is "1-1" "onto"; theorem :: AUTALG_1:15 for A, B be non-empty ManySortedSet of I for F be ManySortedFunction of A, B st F is "1-1" "onto" holds (F"")"" = F; theorem :: AUTALG_1:16 for F, G being Function-yielding Function st F is "1-1" & G is "1-1" holds G ** F is "1-1"; theorem :: AUTALG_1:17 for B, C be non-empty ManySortedSet of I for F be ManySortedFunction of A, B for G be ManySortedFunction of B, C st F is "onto" & G is "onto" holds G ** F is "onto"; theorem :: AUTALG_1:18 for A, B, C be non-empty ManySortedSet of I for F be ManySortedFunction of A, B for G be ManySortedFunction of B, C st F is "1-1" "onto" & G is "1-1" "onto" holds (G ** F)"" = (F"") ** (G""); theorem :: AUTALG_1:19 for A, B be non-empty ManySortedSet of I for F be ManySortedFunction of A, B for G be ManySortedFunction of B, A st F is "1-1" & F is "onto" & G ** F = id A holds G = F""; begin :: On the Group of Automorphisms of Many Sorted Algebra reserve S for non void non empty ManySortedSign, U1, U2 for non-empty MSAlgebra over S; definition let I, A, B; func MSFuncs (A, B) -> ManySortedSet of I means :: AUTALG_1:def 5 for i be set st i in I holds it.i = Funcs(A.i, B.i); end; canceled; theorem :: AUTALG_1:21 for A, B be ManySortedSet of I st A is_transformable_to B for x be set st x in product MSFuncs(A, B) holds x is ManySortedFunction of A, B; theorem :: AUTALG_1:22 for A, B be ManySortedSet of I st A is_transformable_to B for g be ManySortedFunction of A, B holds g in product MSFuncs(A, B); theorem :: AUTALG_1:23 for A, B be ManySortedSet of I st A is_transformable_to B holds MSFuncs(A, B) is non-empty; definition let I, A, B; assume A is_transformable_to B; mode MSFunctionSet of A, B -> non empty set means :: AUTALG_1:def 6 for x be set st x in it holds x is ManySortedFunction of A, B; end; definition let I, A; cluster MSFuncs(A, A) -> non-empty; end; definition let S, U1, U2; mode MSFunctionSet of U1, U2 is MSFunctionSet of the Sorts of U1, the Sorts of U2; end; definition let I be set; let D be ManySortedSet of I; cluster non empty MSFunctionSet of D, D; end; definition let I be set; let D be ManySortedSet of I; let A be non empty MSFunctionSet of D, D; redefine mode Element of A -> ManySortedFunction of D, D; end; theorem :: AUTALG_1:24 id A is "onto"; theorem :: AUTALG_1:25 id A is "1-1"; canceled; theorem :: AUTALG_1:27 id the Sorts of U1 in product MSFuncs (the Sorts of U1, the Sorts of U1); definition let S, U1; func MSAAut U1 -> MSFunctionSet of the Sorts of U1, the Sorts of U1 means :: AUTALG_1:def 7 (for f be Element of it holds f is ManySortedFunction of U1, U1) & for h be ManySortedFunction of U1, U1 holds h in it iff h is_isomorphism U1, U1; end; canceled; theorem :: AUTALG_1:29 for f be Element of MSAAut U1 holds f in product MSFuncs (the Sorts of U1, the Sorts of U1); theorem :: AUTALG_1:30 MSAAut U1 c= product MSFuncs (the Sorts of U1, the Sorts of U1); theorem :: AUTALG_1:31 id the Sorts of U1 in MSAAut U1; theorem :: AUTALG_1:32 for f be Element of MSAAut U1 holds f"" in MSAAut U1; theorem :: AUTALG_1:33 for f1, f2 be Element of MSAAut U1 holds f1 ** f2 in MSAAut U1; theorem :: AUTALG_1:34 for F be ManySortedFunction of MSAlg UA, MSAlg UA for f be Element of UAAut UA st F = {0} --> f holds F in MSAAut MSAlg UA; definition let S, U1; func MSAAutComp U1 -> BinOp of MSAAut U1 means :: AUTALG_1:def 8 for x, y be Element of MSAAut U1 holds it.(x, y) = y ** x; end; definition let S, U1; func MSAAutGroup U1 -> Group equals :: AUTALG_1:def 9 HGrStr (# MSAAut U1, MSAAutComp U1 #); end; definition let S, U1; cluster MSAAutGroup U1 -> strict; end; theorem :: AUTALG_1:35 for x, y be Element of MSAAutGroup U1 for f, g be Element of MSAAut U1 st x = f & y = g holds x * y = g ** f; theorem :: AUTALG_1:36 id the Sorts of U1 = 1.MSAAutGroup U1; theorem :: AUTALG_1:37 for f be Element of MSAAut U1 for g be Element of MSAAutGroup U1 st f = g holds f"" = g"; begin :: On the Relationship of Automorphisms of 1-Sorted and Many Sorted Algebras theorem :: AUTALG_1:38 for UA1, UA2 be Universal_Algebra st UA1, UA2 are_similar for F be ManySortedFunction of MSAlg UA1, (MSAlg UA2 Over MSSign UA1) holds F.0 is Function of UA1, UA2; theorem :: AUTALG_1:39 for f be Element of UAAut UA holds {0} --> f is ManySortedFunction of MSAlg UA, MSAlg UA; theorem :: AUTALG_1:40 for h be Function st (dom h = UAAut UA & for x be set st x in UAAut UA holds h.x = {0} --> x) holds h is Homomorphism of UAAutGroup UA, MSAAutGroup (MSAlg UA); theorem :: AUTALG_1:41 for h be Homomorphism of UAAutGroup UA, MSAAutGroup (MSAlg UA) st for x be set st x in UAAut UA holds h.x = {0} --> x holds h is_isomorphism; theorem :: AUTALG_1:42 UAAutGroup UA, MSAAutGroup (MSAlg UA) are_isomorphic;

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