:: On the Lattice of Subalgebras of a Universal Algebra
:: by Miros{\l}aw Jan Paszek
::
:: Received May 23, 1995
:: Copyright (c) 1995-2011 Association of Mizar Users

begin

definition

let U0 be Universal_Algebra;
mode SubAlgebra-Family of U0 -> set means :Def1: :: UNIALG_3:def 1
for U1 being set st U1 in it holds
U1 is SubAlgebra of U0;
existence
ex b₁ being set st
for U1 being set st U1 in b₁ holds
U1 is SubAlgebra of U0

proof end;

end;

:: deftheorem Def1 defines SubAlgebra-Family UNIALG_3:def 1 :
for U0 being Universal_Algebra
for b₂ being set holds
( b₂ is SubAlgebra-Family of U0 iff for U1 being set st U1 in b₂ holds
U1 is SubAlgebra of U0 );

registration

let U0 be Universal_Algebra;
cluster non empty SubAlgebra-Family of U0;
existence
not for b₁ being SubAlgebra-Family of U0 holds b₁ is empty

proof end;

end;

definition

let U0 be Universal_Algebra;
:: original: Sub
redefine func Sub U0 -> non empty SubAlgebra-Family of U0;
coherence
Sub U0 is non empty SubAlgebra-Family of U0

proof end;

let U00 be non empty SubAlgebra-Family of U0;
:: original: Element
redefine mode Element of U00 -> SubAlgebra of U0;
coherence
for b₁ being Element of U00 holds b₁ is SubAlgebra of U0 by Def1;

end;

definition

let U0 be Universal_Algebra;
let u be Element of Sub U0;
func carr u -> Subset of U0 means :Def2: :: UNIALG_3:def 2
ex U1 being SubAlgebra of U0 st
( u = U1 & it = the carrier of U1 );
existence
ex b₁ being Subset of U0 ex U1 being SubAlgebra of U0 st
( u = U1 & b₁ = the carrier of U1 )

proof end;

uniqueness
for b₁, b₂ being Subset of U0 st ex U1 being SubAlgebra of U0 st
( u = U1 & b₁ = the carrier of U1 ) & ex U1 being SubAlgebra of U0 st
( u = U1 & b₂ = the carrier of U1 ) holds
b₁ = b₂ ;

end;

:: deftheorem Def2 defines carr UNIALG_3:def 2 :
for U0 being Universal_Algebra
for u being Element of Sub U0
for b₃ being Subset of U0 holds
( b₃ = carr u iff ex U1 being SubAlgebra of U0 st
( u = U1 & b₃ = the carrier of U1 ) );

definition

let U0 be Universal_Algebra;
func Carr U0 -> Function of (Sub U0),(bool the carrier of U0) means :Def3: :: UNIALG_3:def 3
for u being Element of Sub U0 holds it . u = carr u;
existence
ex b₁ being Function of (Sub U0),(bool the carrier of U0) st
for u being Element of Sub U0 holds b₁ . u = carr u

proof end;

uniqueness
for b₁, b₂ being Function of (Sub U0),(bool the carrier of U0) st ( for u being Element of Sub U0 holds b₁ . u = carr u ) & ( for u being Element of Sub U0 holds b₂ . u = carr u ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines Carr UNIALG_3:def 3 :
for U0 being Universal_Algebra
for b₂ being Function of (Sub U0),(bool the carrier of U0) holds
( b₂ = Carr U0 iff for u being Element of Sub U0 holds b₂ . u = carr u );

theorem Th1: :: UNIALG_3:1

for U0 being Universal_Algebra
for u being set holds
( u in Sub U0 iff ex U1 being strict SubAlgebra of U0 st u = U1 )

proof end;

theorem :: UNIALG_3:2

for U0 being Universal_Algebra
for H being non empty Subset of U0
for o being operation of U0 st arity o = 0 holds
( H is_closed_on o iff o . {} in H )

proof end;

theorem Th3: :: UNIALG_3:3

for U0 being Universal_Algebra
for U1 being SubAlgebra of U0 holds the carrier of U1 c= the carrier of U0

proof end;

theorem :: UNIALG_3:4

for U0 being Universal_Algebra
for H being non empty Subset of U0
for o being operation of U0 st H is_closed_on o & arity o = 0 holds
o /. H = o

proof end;

theorem :: UNIALG_3:5

for U0 being Universal_Algebra holds Constants U0 = { (o . {}) where o is operation of U0 : arity o = 0 }

proof end;

theorem Th6: :: UNIALG_3:6

for U0 being with_const_op Universal_Algebra
for U1 being SubAlgebra of U0 holds Constants U0 = Constants U1

proof end;

registration

let U0 be with_const_op Universal_Algebra;
cluster -> with_const_op SubAlgebra of U0;
coherence
for b₁ being SubAlgebra of U0 holds b₁ is with_const_op

proof end;

end;

theorem :: UNIALG_3:7

for U0 being with_const_op Universal_Algebra
for U1, U2 being SubAlgebra of U0 holds Constants U1 = Constants U2

proof end;

definition

let U0 be Universal_Algebra;
redefine func Carr U0 means :Def4: :: UNIALG_3:def 4
for u being Element of Sub U0
for U1 being SubAlgebra of U0 st u = U1 holds
it . u = the carrier of U1;
compatibility
for b₁ being Function of (Sub U0),(bool the carrier of U0) holds
( b₁ = Carr U0 iff for u being Element of Sub U0
for U1 being SubAlgebra of U0 st u = U1 holds
b₁ . u = the carrier of U1 )

proof end;

end;

:: deftheorem Def4 defines Carr UNIALG_3:def 4 :
for U0 being Universal_Algebra
for b₂ being Function of (Sub U0),(bool the carrier of U0) holds
( b₂ = Carr U0 iff for u being Element of Sub U0
for U1 being SubAlgebra of U0 st u = U1 holds
b₂ . u = the carrier of U1 );

theorem :: UNIALG_3:8

for U0 being Universal_Algebra
for H being strict SubAlgebra of U0
for u being Element of U0 holds
( u in (Carr U0) . H iff u in H )

proof end;

theorem Th9: :: UNIALG_3:9

for U0 being Universal_Algebra
for H being non empty Subset of (Sub U0) holds not (Carr U0) .: H is empty

proof end;

theorem :: UNIALG_3:10

for U0 being with_const_op Universal_Algebra
for U1 being strict SubAlgebra of U0 holds Constants U0 c= (Carr U0) . U1

proof end;

theorem Th11: :: UNIALG_3:11

for U0 being with_const_op Universal_Algebra
for U1 being SubAlgebra of U0
for a being set st a is Element of Constants U0 holds
a in the carrier of U1

proof end;

theorem Th12: :: UNIALG_3:12

for U0 being with_const_op Universal_Algebra
for H being non empty Subset of (Sub U0) holds meet ((Carr U0) .: H) is non empty Subset of U0

proof end;

theorem :: UNIALG_3:13

for U0 being with_const_op Universal_Algebra holds the carrier of (UnSubAlLattice U0) = Sub U0 ;

theorem Th14: :: UNIALG_3:14

for U0 being with_const_op Universal_Algebra
for H being non empty Subset of (Sub U0)
for S being non empty Subset of U0 st S = meet ((Carr U0) .: H) holds
S is opers_closed

proof end;

definition

let U0 be strict with_const_op Universal_Algebra;
let H be non empty Subset of (Sub U0);
func meet H -> strict SubAlgebra of U0 means :Def5: :: UNIALG_3:def 5
the carrier of it = meet ((Carr U0) .: H);
existence
ex b₁ being strict SubAlgebra of U0 st the carrier of b₁ = meet ((Carr U0) .: H)

proof end;

uniqueness
for b₁, b₂ being strict SubAlgebra of U0 st the carrier of b₁ = meet ((Carr U0) .: H) & the carrier of b₂ = meet ((Carr U0) .: H) holds
b₁ = b₂ by UNIALG_2:15;

end;

:: deftheorem Def5 defines meet UNIALG_3:def 5 :
for U0 being strict with_const_op Universal_Algebra
for H being non empty Subset of (Sub U0)
for b₃ being strict SubAlgebra of U0 holds
( b₃ = meet H iff the carrier of b₃ = meet ((Carr U0) .: H) );

theorem :: UNIALG_3:15

canceled;

theorem :: UNIALG_3:16

canceled;

theorem Th17: :: UNIALG_3:17

for U0 being with_const_op Universal_Algebra
for l1, l2 being Element of (UnSubAlLattice U0)
for U1, U2 being strict SubAlgebra of U0 st l1 = U1 & l2 = U2 holds
( l1 [= l2 iff the carrier of U1 c= the carrier of U2 )

proof end;

theorem :: UNIALG_3:18

for U0 being with_const_op Universal_Algebra
for l1, l2 being Element of (UnSubAlLattice U0)
for U1, U2 being strict SubAlgebra of U0 st l1 = U1 & l2 = U2 holds
( l1 [= l2 iff U1 is SubAlgebra of U2 )

proof end;

theorem Th19: :: UNIALG_3:19

for U0 being strict with_const_op Universal_Algebra holds UnSubAlLattice U0 is bounded

proof end;

registration

let U0 be strict with_const_op Universal_Algebra;
cluster UnSubAlLattice U0 -> bounded ;
coherence
UnSubAlLattice U0 is bounded by Th19;

end;

theorem Th20: :: UNIALG_3:20

for U0 being strict with_const_op Universal_Algebra
for U1 being strict SubAlgebra of U0 holds (GenUnivAlg (Constants U0)) /\ U1 = GenUnivAlg (Constants U0)

proof end;

theorem :: UNIALG_3:21

for U0 being strict with_const_op Universal_Algebra holds Bottom (UnSubAlLattice U0) = GenUnivAlg (Constants U0)

proof end;

theorem Th22: :: UNIALG_3:22

for U0 being strict with_const_op Universal_Algebra
for U1 being SubAlgebra of U0
for H being Subset of U0 st H = the carrier of U0 holds
(GenUnivAlg H) "\/" U1 = GenUnivAlg H

proof end;

theorem Th23: :: UNIALG_3:23

for U0 being strict with_const_op Universal_Algebra
for H being Subset of U0 st H = the carrier of U0 holds
Top (UnSubAlLattice U0) = GenUnivAlg H

proof end;

theorem :: UNIALG_3:24

for U0 being strict with_const_op Universal_Algebra holds Top (UnSubAlLattice U0) = U0

proof end;

theorem :: UNIALG_3:25

for U0 being strict with_const_op Universal_Algebra holds UnSubAlLattice U0 is complete

proof end;

definition

let U01, U02 be with_const_op Universal_Algebra;
let F be Function of the carrier of U01, the carrier of U02;
func FuncLatt F -> Function of the carrier of (UnSubAlLattice U01), the carrier of (UnSubAlLattice U02) means :Def6: :: UNIALG_3:def 6
for U1 being strict SubAlgebra of U01
for H being Subset of U02 st H = F .: the carrier of U1 holds
it . U1 = GenUnivAlg H;
existence
ex b₁ being Function of the carrier of (UnSubAlLattice U01), the carrier of (UnSubAlLattice U02) st
for U1 being strict SubAlgebra of U01
for H being Subset of U02 st H = F .: the carrier of U1 holds
b₁ . U1 = GenUnivAlg H

proof end;

uniqueness
for b₁, b₂ being Function of the carrier of (UnSubAlLattice U01), the carrier of (UnSubAlLattice U02) st ( for U1 being strict SubAlgebra of U01
for H being Subset of U02 st H = F .: the carrier of U1 holds
b₁ . U1 = GenUnivAlg H ) & ( for U1 being strict SubAlgebra of U01
for H being Subset of U02 st H = F .: the carrier of U1 holds
b₂ . U1 = GenUnivAlg H ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines FuncLatt UNIALG_3:def 6 :
for U01, U02 being with_const_op Universal_Algebra
for F being Function of the carrier of U01, the carrier of U02
for b₄ being Function of the carrier of (UnSubAlLattice U01), the carrier of (UnSubAlLattice U02) holds
( b₄ = FuncLatt F iff for U1 being strict SubAlgebra of U01
for H being Subset of U02 st H = F .: the carrier of U1 holds
b₄ . U1 = GenUnivAlg H );

theorem :: UNIALG_3:26

for U0 being strict with_const_op Universal_Algebra
for F being Function of the carrier of U0, the carrier of U0 st F = id the carrier of U0 holds
FuncLatt F = id the carrier of (UnSubAlLattice U0)

proof end;