:: Congruences and Quotient Algebras of {BCI}-algebras
:: by Yuzhong Ding and Zhiyong Pang
::
:: Received August 28, 2007
:: Copyright (c) 2007-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies BCIALG_1, CARD_FIL, SUBSET_1, FUNCT_1, NUMBERS, STRUCT_0, POWER,
CARD_1, ARYTM_3, XBOOLE_0, XXREAL_0, FUNCOP_1, NAT_1, SUPINF_2, RELAT_1,
CHORD, WAYBEL15, GROUP_4, GROUP_1, ALG_1, EQREL_1, PARTFUN1, RELAT_2,
ZFMISC_1, RCOMP_1, TARSKI, BINOP_1, GROUP_6, BCIALG_2, XCMPLX_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, RELAT_1,
FUNCT_1, XCMPLX_0, NAT_1, FUNCT_2, XXREAL_0, FUNCOP_1, BINOP_1, STRUCT_0,
BCIALG_1, RELAT_2, PARTFUN1, EQREL_1, ALG_1;
constructors BCIALG_1, BINOP_1, NAT_1, BINARITH, EQREL_1, RELSET_1, XREAL_0,
NUMBERS, FUNCOP_1;
registrations BCIALG_1, SUBSET_1, NAT_1, RELSET_1, STRUCT_0, PARTFUN1,
ORDINAL1, XREAL_0, CARD_1;
requirements NUMERALS, ARITHM, SUBSET, BOOLE;
begin :: Basic Properties of The Element P16-P19
reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of the carrier of X;
reserve j,i,k,n,m for Nat;
:: x*y to_power n = x*y|^n
:: n=0: it=x; n=1:it =x*y; n=2:it=(x*y)*y...
definition
let X,x,y;
let n be Nat;
func (x,y) to_power n -> Element of X means
:: BCIALG_2:def 1
ex f st it = f.n & f.0 = x & for j being Nat st j positive minimal;
end;
theorem :: BCIALG_2:23
a is minimal iff for x holds a\x=x`\a`;
theorem :: BCIALG_2:24
x` is minimal iff for y holds y<=x implies x` = y`;
theorem :: BCIALG_2:25
x` is minimal iff for y,z holds (((x\z)\(y\z))`)` = y`\x`;
theorem :: BCIALG_2:26
0.X is maximal implies for a holds a is minimal;
theorem :: BCIALG_2:27
(ex x st x is greatest) implies for a holds a is positive;
theorem :: BCIALG_2:28
x\(x``) is positive Element of X;
theorem :: BCIALG_2:29
a is minimal iff a`` = a;
theorem :: BCIALG_2:30
a is minimal iff ex x st a=x`;
:: p38
definition
let X,x;
attr x is nilpotent means
:: BCIALG_2:def 6
ex k being non zero Nat st (0.X ,x) to_power k = 0.X;
end;
definition
let X;
attr X is nilpotent means
:: BCIALG_2:def 7
for x being Element of X holds x is nilpotent;
end;
definition
let X,x;
assume
x is nilpotent;
func ord x -> non zero Nat means
:: BCIALG_2:def 8
(0.X,x) to_power it = 0.
X & for m being Nat st (0.X,x) to_power m=0.X & m <> 0 holds it <= m;
end;
registration
let X;
cluster 0.X -> nilpotent;
end;
theorem :: BCIALG_2:31 :: P39
x is positive Element of X iff x is nilpotent & ord x=1;
theorem :: BCIALG_2:32
X is BCK-algebra iff for x holds ord x=1 & x is nilpotent;
theorem :: BCIALG_2:33
(0.X,x`) to_power n is minimal;
theorem :: BCIALG_2:34
x is nilpotent implies ord x = ord (x`);
begin :: Congruence and Quotient Algebras P58-65
definition
let X be BCI-algebra;
mode Congruence of X -> Equivalence_Relation of X means
:: BCIALG_2:def 9
for x,y,u,v
being Element of X st [x,y] in it & [u,v] in it holds [x\u,y\v] in it;
end;
:: Left Congruence
definition
let X be BCI-algebra;
mode L-congruence of X -> Equivalence_Relation of X means
:: BCIALG_2:def 10
for x,y
being Element of X st [x,y] in it holds for u being Element of X holds [u\x,u\y
] in it;
end;
:: Right Congruence
definition
let X be BCI-algebra;
mode R-congruence of X -> Equivalence_Relation of X means
:: BCIALG_2:def 11
for x,y
being Element of X st [x,y] in it holds for u being Element of X holds [x\u,y\u
] in it;
end;
::Ideal Congruence
definition
let X be BCI-algebra,A be Ideal of X;
mode I-congruence of X,A -> Relation of X means
:: BCIALG_2:def 12
for x,y being Element of X holds [x,y] in it iff x\y in A & y\x in A;
end;
registration
let X be BCI-algebra, A be Ideal of X;
cluster -> total symmetric transitive for I-congruence of X,A;
end;
definition
let X be BCI-algebra;
func IConSet(X) -> set means
:: BCIALG_2:def 13
for A1 being set holds A1 in it iff ex I being
Ideal of X st A1 is I-congruence of X,I;
end;
definition
let X be BCI-algebra;
func ConSet(X) -> set equals
:: BCIALG_2:def 14
the set of all R where R is Congruence of X;
func LConSet(X) -> set equals
:: BCIALG_2:def 15
the set of all R where R is L-congruence of X;
func RConSet(X) -> set equals
:: BCIALG_2:def 16
the set of all R where R is R-congruence of X;
end;
reserve R for Equivalence_Relation of X;
reserve RI for I-congruence of X,I;
reserve E for Congruence of X;
reserve RC for R-congruence of X;
reserve LC for L-congruence of X;
:: huang-P58:P1.5.1
theorem :: BCIALG_2:35
for X,E holds Class(E,0.X) is closed Ideal of X;
theorem :: BCIALG_2:36
R is Congruence of X iff R is R-congruence of X&R is L-congruence of X;
theorem :: BCIALG_2:37
RI is Congruence of X;
definition
let X be BCI-algebra, I be Ideal of X;
redefine mode I-congruence of X,I -> Congruence of X;
end;
theorem :: BCIALG_2:38
Class(RI,0.X) c= I;
theorem :: BCIALG_2:39
I is closed iff I = Class(RI,0.X);
theorem :: BCIALG_2:40
[x,y] in E implies x\y in Class(E,0.X) & y\x in Class(E,0.X);
theorem :: BCIALG_2:41
for A,I being Ideal of X,IA being I-congruence of X,A, II being
I-congruence of X,I st Class(IA,0.X)=Class(II,0.X) holds IA = II;
theorem :: BCIALG_2:42
[x,y] in E & u in Class(E,0.X) implies [x,(y,u)to_power k] in E;
theorem :: BCIALG_2:43
(for X,x,y holds ex i,j,m,n st ((x,x\y) to_power i,y\x) to_power j = (
(y,y\x)to_power m,x\y) to_power n) implies for E,I st I=Class(E,0.X) holds E is
I-congruence of X,I;
theorem :: BCIALG_2:44
IConSet(X) c= ConSet(X);
theorem :: BCIALG_2:45
ConSet(X) c= LConSet(X);
theorem :: BCIALG_2:46
ConSet(X) c= RConSet(X);
theorem :: BCIALG_2:47
ConSet(X) = LConSet(X)/\RConSet(X);
theorem :: BCIALG_2:48
(for LC holds LC is I-congruence of X,I) implies E = RI;
theorem :: BCIALG_2:49
(for RC holds RC is I-congruence of X,I) implies E = RI;
theorem :: BCIALG_2:50
Class(LC,0.X) is closed Ideal of X;
:: Quotient Algebras
reserve E for Congruence of X;
reserve RI for I-congruence of X,I;
registration
let X,E;
cluster Class E -> non empty;
end;
definition
let X,E;
func EqClaOp E -> BinOp of Class E means
:: BCIALG_2:def 17
for W1,W2 being Element of
Class(E) for x,y st W1=Class(E,x) & W2=Class(E,y) holds it.(W1,W2)=Class(E,x\y)
;
end;
definition
let X,E;
func zeroEqC(E) -> Element of Class E equals
:: BCIALG_2:def 18
Class(E,0.X);
end;
::Quotient Algebras
definition
let X,E;
func X./.E -> BCIStr_0 equals
:: BCIALG_2:def 19
BCIStr_0(#Class E,EqClaOp E,zeroEqC(E)#);
end;
registration
let X;
let E be Congruence of X;
cluster X./.E -> non empty;
end;
reserve W1,W2 for Element of Class E;
definition
let X,E,W1,W2;
func W1\W2 -> Element of Class E equals
:: BCIALG_2:def 20
(EqClaOp E).(W1,W2);
end;
theorem :: BCIALG_2:51
X./.RI is BCI-algebra;
registration
let X,I,RI;
cluster X./.RI -> strict being_B being_C being_I being_BCI-4;
end;
theorem :: BCIALG_2:52
for X,I st I = BCK-part(X) holds for RI being I-congruence of X,I
holds X./.RI is p-Semisimple BCI-algebra;