:: MSUHOM_1 semantic presentation

theorem Th1: :: MSUHOM_1:1

for f, g being Function
for C being set st rng f c= C holds
(g | C) * f = g * f

proof end;

theorem Th2: :: MSUHOM_1:2

for I being set
for C being Subset of I holds C * c= I *

proof end;

theorem :: MSUHOM_1:3

for f being Function
for C being set st f is Function-yielding holds
f | C is Function-yielding

proof end;

theorem Th4: :: MSUHOM_1:4

for I being set
for C being Subset of I
for M being ManySortedSet of I holds (M | C) # = (M # ) | (C * )

proof end;

definition

let A be non empty set ;
let n be Nat;
let a be Element of A;
:: original: |->
redefine func n |-> a -> FinSequence of A;
coherence by FINSEQ_2:77;

end;

definition

let S, S' be non empty ManySortedSign ;
pred S <= S' means :Def1: :: MSUHOM_1:def 1
( the carrier of S c= the carrier of S' & the OperSymbols of S c= the OperSymbols of S' & the Arity of S' | the OperSymbols of S = the Arity of S & the ResultSort of S' | the OperSymbols of S = the ResultSort of S );
reflexivity

proof end;

end;

:: deftheorem Def1 defines <= MSUHOM_1:def 1 :

theorem :: MSUHOM_1:5

for S, S', S'' being non empty ManySortedSign st S <= S' & S' <= S'' holds
S <= S''

proof end;

theorem :: MSUHOM_1:6

for S, S' being non empty strict ManySortedSign st S <= S' & S' <= S holds
S = S'

proof end;

theorem :: MSUHOM_1:7

for n being Nat
for A being non empty set
for g being Function
for a being Element of A
for k being Nat st 1 <= k & k <= n holds
(a .--> g) . ((n |-> a) /. k) = g

proof end;

theorem Th8: :: MSUHOM_1:8

for I being set
for I0 being Subset of I
for A, B being ManySortedSet of I
for F being ManySortedFunction of A,B
for A0, B0 being ManySortedSet of I0 st A0 = A | I0 & B0 = B | I0 holds
F | I0 is ManySortedFunction of A0,B0

proof end;

definition

let S, S' be non empty strict non void ManySortedSign ;
let A be strict non-empty MSAlgebra of S';
assume A1: S <= S' ;
func A Over S -> strict non-empty MSAlgebra of S means :Def2: :: MSUHOM_1:def 2
( the Sorts of it = the Sorts of A | the carrier of S & the Charact of it = the Charact of A | the OperSymbols of S );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def2 defines Over MSUHOM_1:def 2 :

theorem Th9: :: MSUHOM_1:9

for S being non empty strict non void ManySortedSign
for A being strict non-empty MSAlgebra of S holds A = A Over S

proof end;

theorem Th10: :: MSUHOM_1:10

for U1, U2 being Universal_Algebra st U1,U2 are_similar holds
MSSign U1 = MSSign U2

proof end;

definition

let U1, U2 be Universal_Algebra;
let h be Function of U1,U2;
assume A1: MSSign U1 = MSSign U2 ;
func MSAlg h -> ManySortedFunction of (MSAlg U1),((MSAlg U2) Over (MSSign U1)) equals :Def3: :: MSUHOM_1:def 3
{0} --> h;
coherence

proof end;

end;

:: deftheorem Def3 defines MSAlg MSUHOM_1:def 3 :

theorem Th11: :: MSUHOM_1:11

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar holds
for o being OperSymbol of (MSSign U1) holds (MSAlg h) . (the_result_sort_of o) = h

proof end;

theorem Th12: :: MSUHOM_1:12

for U1 being Universal_Algebra
for o being OperSymbol of (MSSign U1) holds Den o,(MSAlg U1) = the charact of U1 . o

proof end;

Lm1: for U1 being Universal_Algebra holds dom (signature U1) = dom the charact of U1

proof end;

theorem Th13: :: MSUHOM_1:13

for U1 being Universal_Algebra
for o being OperSymbol of (MSSign U1) holds Den o,(MSAlg U1) is operation of U1

proof end;

Lm2: for U1, U2 being Universal_Algebra st U1,U2 are_similar holds
for o being OperSymbol of (MSSign U1) holds Den o,((MSAlg U2) Over (MSSign U1)) is operation of U2

proof end;

theorem Th14: :: MSUHOM_1:14

for U1 being Universal_Algebra
for o being OperSymbol of (MSSign U1)
for y being Element of Args o,(MSAlg U1) holds y is FinSequence of the carrier of U1

proof end;

theorem Th15: :: MSUHOM_1:15

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar holds
for o being OperSymbol of (MSSign U1)
for y being Element of Args o,(MSAlg U1) holds (MSAlg h) # y = h * y

proof end;

theorem Th16: :: MSUHOM_1:16

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st h is_homomorphism U1,U2 holds
MSAlg h is_homomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1)

proof end;

Lm3: for U1 being Universal_Algebra
for n being Nat st n in dom the charact of U1 holds
n is OperSymbol of (MSSign U1)

proof end;

theorem Th17: :: MSUHOM_1:17

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar holds
MSAlg h is ManySortedSet of {0}

proof end;

theorem Th18: :: MSUHOM_1:18

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st h is_epimorphism U1,U2 holds
MSAlg h is_epimorphism MSAlg U1,(MSAlg U2) Over (MSSign U1)

proof end;

theorem Th19: :: MSUHOM_1:19

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st h is_monomorphism U1,U2 holds
MSAlg h is_monomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1)

proof end;

theorem :: MSUHOM_1:20

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st h is_isomorphism U1,U2 holds
MSAlg h is_isomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1)

proof end;

theorem Th21: :: MSUHOM_1:21

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar & MSAlg h is_homomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1) holds
h is_homomorphism U1,U2

proof end;

theorem Th22: :: MSUHOM_1:22

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar & MSAlg h is_epimorphism MSAlg U1,(MSAlg U2) Over (MSSign U1) holds
h is_epimorphism U1,U2

proof end;

theorem Th23: :: MSUHOM_1:23

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar & MSAlg h is_monomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1) holds
h is_monomorphism U1,U2

proof end;

theorem :: MSUHOM_1:24

for U1, U2 being Universal_Algebra
for h being Function of U1,U2 st U1,U2 are_similar & MSAlg h is_isomorphism MSAlg U1,(MSAlg U2) Over (MSSign U1) holds
h is_isomorphism U1,U2

proof end;

theorem :: MSUHOM_1:25

for U1 being Universal_Algebra holds MSAlg (id the carrier of U1) = id the Sorts of (MSAlg U1)

proof end;

theorem :: MSUHOM_1:26

for U1, U2, U3 being Universal_Algebra st U1,U2 are_similar & U2,U3 are_similar holds
for h1 being Function of U1,U2
for h2 being Function of U2,U3 holds (MSAlg h2) ** (MSAlg h1) = MSAlg (h2 * h1)

proof end;