Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

### The abstract of the Mizar article:

### On the Trivial Many Sorted Algebras and Many Sorted Congruences

**by****Artur Kornilowicz**- Received June 11, 1996
- MML identifier: MSUALG_9

- [ Mizar article, MML identifier index ]

environ vocabulary AMI_1, MSUALG_1, PBOOLE, FUNCT_1, PRE_CIRC, CLOSURE2, CAT_4, MSUALG_2, ZF_REFLE, FINSET_1, FINSEQ_1, RELAT_1, QC_LANG1, PRALG_2, CARD_3, RLVECT_2, MSAFREE, PRELAMB, PRALG_1, FUNCT_3, MCART_1, EQREL_1, FUNCOP_1, BOOLE, MSUALG_3, TREES_4, LANG1, BORSUK_1, ALG_1, GROUP_6, WELLORD1, TDGROUP, FINSEQ_4, NATTRA_1, FUNCT_6, MSUALG_4, MSUALG_5, SETFAM_1, FUNCT_4, CANTOR_1, RELAT_2, MSUALG_9; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, NAT_1, RELAT_1, RELAT_2, STRUCT_0, SETFAM_1, RELSET_1, FUNCT_1, EQREL_1, FUNCT_2, FINSEQ_1, LANG1, MCART_1, FINSET_1, CARD_3, FINSEQ_4, CANTOR_1, TREES_4, DTCONSTR, PBOOLE, MSUALG_1, MSUALG_2, PRALG_3, MSUALG_3, MSAFREE, PRALG_2, PRE_CIRC, MSAFREE2, MSUALG_4, AUTALG_1, EXTENS_1, PZFMISC1, MSSUBFAM, CLOSURE1, CLOSURE2, MSUALG_5; constructors AUTALG_1, BINOP_1, CANTOR_1, CLOSURE1, CLOSURE2, EXTENS_1, MSAFREE2, MSUALG_5, PRALG_3, PRE_CIRC, PZFMISC1, FINSEQ_4; clusters CANTOR_1, CLOSURE2, EQREL_1, FINSET_1, MSAFREE, MSSUBFAM, MSUALG_1, MSUALG_2, MSUALG_3, MSUALG_4, MSUALG_5, PRALG_1, PRALG_2, PRE_CIRC, PZFMISC1, RELSET_1, STRUCT_0, ARYTM_3, XBOOLE_0, MEMBERED, PARTFUN1, NUMBERS, ORDINAL2; requirements SUBSET, BOOLE; begin :: Preliminaries reserve a, I for set, S for non empty non void ManySortedSign; scheme MSSExD { I() -> non empty set, P[set,set] }: ex f being ManySortedSet of I() st for i being Element of I() holds P[i,f.i] provided for i being Element of I() ex j being set st P[i,j]; definition let I be set, M be ManySortedSet of I; cluster locally-finite Element of Bool M; end; definition let I be set, M be non-empty ManySortedSet of I; cluster non-empty locally-finite ManySortedSubset of M; end; definition let S be non empty non void ManySortedSign, A be non-empty MSAlgebra over S, o be OperSymbol of S; cluster -> FinSequence-like Element of Args(o,A); end; definition let S be non void non empty ManySortedSign, I be set, s be SortSymbol of S, F be MSAlgebra-Family of I, S; cluster -> Function-like Relation-like Element of (SORTS F).s; end; definition let S be non void non empty ManySortedSign, X be non-empty ManySortedSet of the carrier of S; cluster FreeGen X -> free non-empty; end; definition let S be non void non empty ManySortedSign, X be non-empty ManySortedSet of the carrier of S; cluster FreeMSA X -> free; end; definition let S be non empty non void ManySortedSign, A, B be non-empty MSAlgebra over S; cluster [:A,B:] -> non-empty; end; theorem :: MSUALG_9:1 for X, Y being set, f being Function st a in dom f & f.a in [:X,Y:] holds f.a = [(pr1 f).a, (pr2 f).a]; theorem :: MSUALG_9:2 for X being non empty set, Y being set, f being Function of X, {Y} holds rng f = {Y}; theorem :: MSUALG_9:3 for A being non empty finite set ex f being Function of NAT, A st rng f = A; theorem :: MSUALG_9:4 Class(nabla I) c= {I}; theorem :: MSUALG_9:5 for I being non empty set holds Class(nabla I) = {I}; theorem :: MSUALG_9:6 ex A being ManySortedSet of I st {A} = I --> {a}; theorem :: MSUALG_9:7 for A being ManySortedSet of I ex B being non-empty ManySortedSet of I st A c= B; theorem :: MSUALG_9:8 for M being non-empty ManySortedSet of I for B being locally-finite ManySortedSubset of M ex C being non-empty locally-finite ManySortedSubset of M st B c= C; theorem :: MSUALG_9:9 for A, B being ManySortedSet of I for F, G being ManySortedFunction of A, {B} holds F = G; theorem :: MSUALG_9:10 for A being non-empty ManySortedSet of I, B being ManySortedSet of I for F being ManySortedFunction of A, {B} holds F is "onto"; theorem :: MSUALG_9:11 for A being ManySortedSet of I, B being non-empty ManySortedSet of I for F being ManySortedFunction of {A}, B holds F is "1-1"; theorem :: MSUALG_9:12 for X being non-empty ManySortedSet of the carrier of S holds Reverse X is "1-1"; theorem :: MSUALG_9:13 for A being non-empty locally-finite ManySortedSet of I ex F being ManySortedFunction of I --> NAT, A st F is "onto"; theorem :: MSUALG_9:14 for S being non empty ManySortedSign for A being non-empty MSAlgebra over S for f, g being Element of product the Sorts of A st for i being set holds proj(the Sorts of A,i).f = proj(the Sorts of A,i).g holds f = g; theorem :: MSUALG_9:15 for I being non empty set for s being Element of S for A being MSAlgebra-Family of I,S for f, g being Element of product Carrier(A,s) st for a being Element of I holds proj(Carrier(A,s),a).f = proj(Carrier(A,s),a).g holds f = g; theorem :: MSUALG_9:16 for A, B being non-empty MSAlgebra over S for C being non-empty MSSubAlgebra of A for h1 being ManySortedFunction of B, C st h1 is_homomorphism B, C for h2 being ManySortedFunction of B, A st h1 = h2 holds h2 is_homomorphism B, A; theorem :: MSUALG_9:17 for A, B being non-empty MSAlgebra over S for F being ManySortedFunction of A, B st F is_monomorphism A, B holds A, Image F are_isomorphic; theorem :: MSUALG_9:18 for A, B being non-empty MSAlgebra over S for F being ManySortedFunction of A, B st F is "onto" for o being OperSymbol of S for x being Element of Args(o,B) holds ex y being Element of Args(o,A) st F # y = x; theorem :: MSUALG_9:19 for A being non-empty MSAlgebra over S, o being OperSymbol of S for x being Element of Args(o,A) holds Den(o,A).x in (the Sorts of A).(the_result_sort_of o); theorem :: MSUALG_9:20 for A, B, C being non-empty MSAlgebra over S for F1 being ManySortedFunction of A, B for F2 being ManySortedFunction of A, C st F1 is_epimorphism A, B & F2 is_homomorphism A, C for G being ManySortedFunction of B, C st G ** F1 = F2 holds G is_homomorphism B, C; reserve A, M for ManySortedSet of I, B, C for non-empty ManySortedSet of I; definition let I be set; let A be ManySortedSet of I; let B, C be non-empty ManySortedSet of I; let F be ManySortedFunction of A, [|B,C|]; func Mpr1 F -> ManySortedFunction of A, B means :: MSUALG_9:def 1 for i being set st i in I holds it.i = pr1 (F.i); func Mpr2 F -> ManySortedFunction of A, C means :: MSUALG_9:def 2 for i being set st i in I holds it.i = pr2 (F.i); end; theorem :: MSUALG_9:21 for F being ManySortedFunction of A, [| I-->{a} , I-->{a} |] holds Mpr1 F = Mpr2 F; theorem :: MSUALG_9:22 for F being ManySortedFunction of A, [|B,C|] st F is "onto" holds Mpr1 F is "onto"; theorem :: MSUALG_9:23 for F being ManySortedFunction of A, [|B,C|] st F is "onto" holds Mpr2 F is "onto"; theorem :: MSUALG_9:24 for F being ManySortedFunction of A, [|B,C|] st M in doms F holds for i be set st i in I holds (F..M).i = [((Mpr1 F)..M).i, ((Mpr2 F)..M).i]; begin :: On the Trivial Many Sorted Algebras definition let S be non empty ManySortedSign; cluster the Sorts of Trivial_Algebra S -> locally-finite non-empty; end; definition let S be non empty ManySortedSign; cluster Trivial_Algebra S -> locally-finite non-empty; end; theorem :: MSUALG_9:25 for A being non-empty MSAlgebra over S for F being ManySortedFunction of A, Trivial_Algebra S for o being OperSymbol of S for x being Element of Args(o,A) holds (F.the_result_sort_of o).(Den(o,A).x) = 0 & Den(o,Trivial_Algebra S).(F#x) = 0; theorem :: MSUALG_9:26 for A being non-empty MSAlgebra over S for F being ManySortedFunction of A, Trivial_Algebra S holds F is_epimorphism A, Trivial_Algebra S; theorem :: MSUALG_9:27 for A being MSAlgebra over S st ex X being ManySortedSet of the carrier of S st the Sorts of A = {X} holds A, Trivial_Algebra S are_isomorphic; begin :: On the Many Sorted Congruences theorem :: MSUALG_9:28 for A being non-empty MSAlgebra over S for C being MSCongruence of A holds C is ManySortedSubset of [|the Sorts of A, the Sorts of A|]; theorem :: MSUALG_9:29 for A being non-empty MSAlgebra over S for R being Subset of CongrLatt A for F being SubsetFamily of [|the Sorts of A, the Sorts of A|] st R = F holds meet |:F:| is MSCongruence of A; theorem :: MSUALG_9:30 for A being non-empty MSAlgebra over S for C being MSCongruence of A st C = [|the Sorts of A, the Sorts of A|] holds the Sorts of QuotMSAlg (A,C) = {the Sorts of A}; theorem :: MSUALG_9:31 for A, B being non-empty MSAlgebra over S for F being ManySortedFunction of A, B st F is_homomorphism A, B holds MSHomQuot F ** MSNat_Hom(A,MSCng F) = F; theorem :: MSUALG_9:32 for A being non-empty MSAlgebra over S for C being MSCongruence of A for s being SortSymbol of S for a being Element of (the Sorts of QuotMSAlg (A,C)).s ex x being Element of (the Sorts of A).s st a = Class(C,x); theorem :: MSUALG_9:33 for A being MSAlgebra over S for C1, C2 being MSEquivalence-like ManySortedRelation of A st C1 c= C2 for i being Element of S for x, y being Element of (the Sorts of A).i st [x,y] in C1.i holds Class (C1,x) c= Class (C2,y) & (A is non-empty implies Class (C1,y) c= Class (C2,x)); theorem :: MSUALG_9:34 for A being non-empty MSAlgebra over S, C being MSCongruence of A for s being SortSymbol of S, x, y being Element of (the Sorts of A).s holds (MSNat_Hom(A,C)).s.x = (MSNat_Hom(A,C)).s.y iff [x,y] in C.s; theorem :: MSUALG_9:35 for A being non-empty MSAlgebra over S for C1, C2 being MSCongruence of A for G being ManySortedFunction of QuotMSAlg (A,C1), QuotMSAlg (A,C2) st for i being Element of S for x being Element of (the Sorts of QuotMSAlg (A,C1)).i for xx being Element of (the Sorts of A).i st x = Class(C1,xx) holds G.i.x = Class(C2,xx) holds G ** MSNat_Hom(A,C1) = MSNat_Hom(A,C2); theorem :: MSUALG_9:36 for A being non-empty MSAlgebra over S for C1, C2 being MSCongruence of A for G being ManySortedFunction of QuotMSAlg (A,C1), QuotMSAlg (A,C2) st for i being Element of S for x being Element of (the Sorts of QuotMSAlg (A,C1)).i for xx being Element of (the Sorts of A).i st x = Class(C1,xx) holds G.i.x = Class(C2,xx) holds G is_epimorphism QuotMSAlg (A,C1), QuotMSAlg (A,C2); theorem :: MSUALG_9:37 for A, B being non-empty MSAlgebra over S for F being ManySortedFunction of A, B st F is_homomorphism A, B for C1 being MSCongruence of A st C1 c= MSCng F ex H being ManySortedFunction of QuotMSAlg (A,C1), B st H is_homomorphism QuotMSAlg (A,C1), B & F = H ** MSNat_Hom(A,C1);

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