:: ENDALG semantic presentation
definition
let UA be
Universal_Algebra;
func UAEnd c1 -> FUNCTION_DOMAIN of the
carrier of
a1,the
carrier of
a1 means :
Def1:
:: ENDALG:def 1
for
h being
Function of
UA,
UA holds
(
h in it iff
h is_homomorphism UA,
UA );
existence
ex b1 being FUNCTION_DOMAIN of the carrier of UA,the carrier of UA st
for h being Function of UA,UA holds
( h in b1 iff h is_homomorphism UA,UA )
uniqueness
for b1, b2 being FUNCTION_DOMAIN of the carrier of UA,the carrier of UA st ( for h being Function of UA,UA holds
( h in b1 iff h is_homomorphism UA,UA ) ) & ( for h being Function of UA,UA holds
( h in b2 iff h is_homomorphism UA,UA ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines UAEnd ENDALG:def 1 :
theorem Th1: :: ENDALG:1
theorem Th2: :: ENDALG:2
canceled;
theorem Th3: :: ENDALG:3
theorem Th4: :: ENDALG:4
:: deftheorem Def2 defines UAEndComp ENDALG:def 2 :
:: deftheorem Def3 defines UAEndMonoid ENDALG:def 3 :
theorem Th5: :: ENDALG:5
theorem Th6: :: ENDALG:6
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S;
func MSAEnd c2 -> MSFunctionSet of the
Sorts of
a2,the
Sorts of
a2 means :
Def4:
:: ENDALG:def 4
( ( for
f being
Element of
it holds
f is
ManySortedFunction of
U1,
U1 ) & ( for
h being
ManySortedFunction of
U1,
U1 holds
(
h in it iff
h is_homomorphism U1,
U1 ) ) );
existence
ex b1 being MSFunctionSet of the Sorts of U1,the Sorts of U1 st
( ( for f being Element of b1 holds f is ManySortedFunction of U1,U1 ) & ( for h being ManySortedFunction of U1,U1 holds
( h in b1 iff h is_homomorphism U1,U1 ) ) )
uniqueness
for b1, b2 being MSFunctionSet of the Sorts of U1,the Sorts of U1 st ( for f being Element of b1 holds f is ManySortedFunction of U1,U1 ) & ( for h being ManySortedFunction of U1,U1 holds
( h in b1 iff h is_homomorphism U1,U1 ) ) & ( for f being Element of b2 holds f is ManySortedFunction of U1,U1 ) & ( for h being ManySortedFunction of U1,U1 holds
( h in b2 iff h is_homomorphism U1,U1 ) ) holds
b1 = b2
end;
:: deftheorem Def4 defines MSAEnd ENDALG:def 4 :
theorem Th7: :: ENDALG:7
canceled;
theorem Th8: :: ENDALG:8
canceled;
theorem Th9: :: ENDALG:9
theorem Th10: :: ENDALG:10
theorem Th11: :: ENDALG:11
theorem Th12: :: ENDALG:12
:: deftheorem Def5 defines MSAEndComp ENDALG:def 5 :
:: deftheorem Def6 defines MSAEndMonoid ENDALG:def 6 :
theorem Th13: :: ENDALG:13
theorem Th14: :: ENDALG:14
theorem Th15: :: ENDALG:15
canceled;
theorem Th16: :: ENDALG:16
Lemma89:
for UA being Universal_Algebra
for h being Function st dom h = UAEnd UA & ( for x being set st x in UAEnd UA holds
h . x = {0} --> x ) holds
rng h = MSAEnd (MSAlg UA)
:: deftheorem Def7 ENDALG:def 7 :
canceled;
:: deftheorem Def8 defines unity-preserving ENDALG:def 8 :
:: deftheorem Def9 defines is_monomorphism ENDALG:def 9 :
:: deftheorem Def10 defines is_epimorphism ENDALG:def 10 :
:: deftheorem Def11 defines is_isomorphism ENDALG:def 11 :
theorem Th17: :: ENDALG:17
:: deftheorem Def12 defines are_isomorphic ENDALG:def 12 :
theorem Th18: :: ENDALG:18
theorem Th19: :: ENDALG:19
theorem Th20: :: ENDALG:20