ALG_1: Homomorphisms of algebras. Quotient Universal Algebra

:: Homomorphisms of algebras. Quotient Universal Algebra
:: by Ma{\l}gorzata Korolkiewicz
::
:: Received October 12, 1993
:: Copyright (c) 1993-2021 Association of Mizar Users

theorem Th1: :: ALG_1:1

for U1 being Universal_Algebra
for B being non empty Subset of U1 st B = the carrier of U1 holds
Opers (U1,B) = the charact of U1

proof end;

theorem :: ALG_1:2

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 holds f * (<*> the carrier of U1) = <*> the carrier of U2 ;

theorem Th3: :: ALG_1:3

for U1 being Universal_Algebra
for a being FinSequence of U1 holds (id the carrier of U1) * a = a

proof end;

theorem Th4: :: ALG_1:4

for U1, U2, U3 being Universal_Algebra
for h1 being Function of U1,U2
for h2 being Function of U2,U3
for a being FinSequence of U1 holds h2 * (h1 * a) = (h2 * h1) * a

proof end;

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;

pred f is_homomorphism means :: ALG_1:def 1
( U1,U2 are_similar & ( for n being Nat st n in dom the charact of U1 holds
for o1 being operation of U1
for o2 being operation of U2 st o1 = the charact of U1 . n & o2 = the charact of U2 . n holds
for x being FinSequence of U1 st x in dom o1 holds
f . (o1 . x) = o2 . (f * x) ) );

end;

:: deftheorem defines is_homomorphism ALG_1:def 1 :

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;

pred f is_monomorphism means :: ALG_1:def 2
( f is_homomorphism & f is one-to-one );

pred f is_epimorphism means :: ALG_1:def 3
( f is_homomorphism & rng f = the carrier of U2 );

end;

:: deftheorem defines is_monomorphism ALG_1:def 2 :

:: deftheorem defines is_epimorphism ALG_1:def 3 :

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;

pred f is_isomorphism means :: ALG_1:def 4
( f is_monomorphism & f is_epimorphism );

end;

:: deftheorem defines is_isomorphism ALG_1:def 4 :

definition

let U1, U2 be Universal_Algebra;

pred U1,U2 are_isomorphic means :: ALG_1:def 5
ex f being Function of U1,U2 st f is_isomorphism ;

end;

:: deftheorem defines are_isomorphic ALG_1:def 5 :

theorem Th5: :: ALG_1:5

for U1 being Universal_Algebra holds id the carrier of U1 is_homomorphism

proof end;

theorem Th6: :: ALG_1:6

for U1, U2, U3 being Universal_Algebra
for h1 being Function of U1,U2
for h2 being Function of U2,U3 st h1 is_homomorphism & h2 is_homomorphism holds
h2 * h1 is_homomorphism

proof end;

theorem Th7: :: ALG_1:7

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 holds
( f is_isomorphism iff ( f is_homomorphism & rng f = the carrier of U2 & f is one-to-one ) )

proof end;

theorem Th8: :: ALG_1:8

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 st f is_isomorphism holds
( dom f = the carrier of U1 & rng f = the carrier of U2 )

proof end;

theorem Th9: :: ALG_1:9

for U1, U2 being Universal_Algebra
for h being Function of U1,U2
for h1 being Function of U2,U1 st h is_isomorphism & h1 = h " holds
h1 is_homomorphism

proof end;

theorem Th10: :: ALG_1:10

for U1, U2 being Universal_Algebra
for h being Function of U1,U2
for h1 being Function of U2,U1 st h is_isomorphism & h1 = h " holds
h1 is_isomorphism

proof end;

theorem Th11: :: ALG_1:11

for U1, U2, U3 being Universal_Algebra
for h being Function of U1,U2
for h1 being Function of U2,U3 st h is_isomorphism & h1 is_isomorphism holds
h1 * h is_isomorphism

proof end;

theorem :: ALG_1:12

for U1 being Universal_Algebra holds U1,U1 are_isomorphic

proof end;

theorem :: ALG_1:13

for U1, U2 being Universal_Algebra st U1,U2 are_isomorphic holds
U2,U1 are_isomorphic

proof end;

theorem :: ALG_1:14

for U1, U2, U3 being Universal_Algebra st U1,U2 are_isomorphic & U2,U3 are_isomorphic holds
U1,U3 are_isomorphic

proof end;

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;
assume A1: f is_homomorphism ;

func Image f -> strict SubAlgebra of U2 means :Def6: :: ALG_1:def 6
the carrier of it = f .: the carrier of U1;

proof end;

proof end;

end;

:: deftheorem Def6 defines Image ALG_1:def 6 :

theorem :: ALG_1:15

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st h is_homomorphism holds
rng h = the carrier of (Image h)

proof end;

theorem :: ALG_1:16

for U1 being Universal_Algebra
for U2 being strict Universal_Algebra
for f being Function of U1,U2 st f is_homomorphism holds
( f is_epimorphism iff Image f = U2 )

proof end;

definition

let U1 be 1-sorted ;
mode Relation of U1 is Relation of the carrier of U1;
mode Equivalence_Relation of U1 is Equivalence_Relation of the carrier of U1;

end;

definition

let U1 be Universal_Algebra;

mode Congruence of U1 -> Equivalence_Relation of U1 means :Def7: :: ALG_1:def 7
for n being Nat
for o1 being operation of U1 st n in dom the charact of U1 & o1 = the charact of U1 . n holds
for x, y being FinSequence of U1 st x in dom o1 & y in dom o1 & [x,y] in ExtendRel it holds
[(o1 . x),(o1 . y)] in it;

proof end;

end;

:: deftheorem Def7 defines Congruence ALG_1:def 7 :

definition

let U1 be Universal_Algebra;
let E be Congruence of U1;
let o be operation of U1;

func QuotOp (o,E) -> non empty homogeneous quasi_total PartFunc of ((Class E) *),(Class E) means :Def8: :: ALG_1:def 8
( dom it = (arity o) -tuples_on (Class E) & ( for y being FinSequence of Class E st y in dom it holds
for x being FinSequence of the carrier of U1 st x is_representatives_FS y holds
it . y = Class (E,(o . x)) ) );

proof end;

proof end;

end;

:: deftheorem Def8 defines QuotOp ALG_1:def 8 :

definition

let U1 be Universal_Algebra;
let E be Congruence of U1;

func QuotOpSeq (U1,E) -> PFuncFinSequence of (Class E) means :Def9: :: ALG_1:def 9
( len it = len the charact of U1 & ( for n being Nat st n in dom it holds
for o1 being operation of U1 st the charact of U1 . n = o1 holds
it . n = QuotOp (o1,E) ) );

proof end;

proof end;

end;

:: deftheorem Def9 defines QuotOpSeq ALG_1:def 9 :

definition

let U1 be Universal_Algebra;
let E be Congruence of U1;

func QuotUnivAlg (U1,E) -> strict Universal_Algebra equals :: ALG_1:def 10
UAStr(# (Class E),(QuotOpSeq (U1,E)) #);

proof end;

end;

:: deftheorem defines QuotUnivAlg ALG_1:def 10 :

definition

let U1 be Universal_Algebra;
let E be Congruence of U1;

func Nat_Hom (U1,E) -> Function of U1,(QuotUnivAlg (U1,E)) means :Def11: :: ALG_1:def 11
for u being Element of U1 holds it . u = Class (E,u);

proof end;

proof end;

end;

:: deftheorem Def11 defines Nat_Hom ALG_1:def 11 :

theorem Th17: :: ALG_1:17

for U1 being Universal_Algebra
for E being Congruence of U1 holds Nat_Hom (U1,E) is_homomorphism

proof end;

theorem :: ALG_1:18

for U1 being Universal_Algebra
for E being Congruence of U1 holds Nat_Hom (U1,E) is_epimorphism

proof end;

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;
assume A1: f is_homomorphism ;

func Cng f -> Congruence of U1 means :Def12: :: ALG_1:def 12
for a, b being Element of U1 holds
( [a,b] in it iff f . a = f . b );

proof end;

proof end;

end;

:: deftheorem Def12 defines Cng ALG_1:def 12 :

definition

let U1, U2 be Universal_Algebra;
let f be Function of U1,U2;
assume A1: f is_homomorphism ;

func HomQuot f -> Function of (QuotUnivAlg (U1,(Cng f))),U2 means :Def13: :: ALG_1:def 13
for a being Element of U1 holds it . (Class ((Cng f),a)) = f . a;

proof end;

proof end;

end;

:: deftheorem Def13 defines HomQuot ALG_1:def 13 :

theorem Th19: :: ALG_1:19

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 st f is_homomorphism holds
( HomQuot f is_homomorphism & HomQuot f is_monomorphism )

proof end;

::

WP:

First isomorphism theorem for universal algebras

theorem Th20: :: ALG_1:20

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 st f is_epimorphism holds
HomQuot f is_isomorphism

proof end;

theorem :: ALG_1:21

for U1, U2 being Universal_Algebra
for f being Function of U1,U2 st f is_epimorphism holds
QuotUnivAlg (U1,(Cng f)),U2 are_isomorphic

proof end;