Fiabilitate si Durabilitate - Fiability & Durability Supplement no 1/ 2013 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 440
PSEUDO ALGEBRAS BOOL APPLIED BL-ALGEBRAS
Asistent univ. Olimpia-Mioara PECINGINA, Univ. C-Tin Brancusi Tg-Jiu
Constantin BOGDAN, Department of Mathematics, University of Craiova, 200585 Craiova, Dolj, Romania, EMAIL tica1234ticabogd@yahoo.com
Abstract: In this article we study Boolean algebras in terms of Hilbert algebras and deductive systems
Key words: Boole algebras
PSEUDO BL-ALGEBRA
Definition 1 A pseudo BL-algebra is an algebra (A , , . , , of type
(2, 2, 2, 2, 2, 0, 0) satisfying the following conditions:
1
(psBL) ( , , , 0,1)A is a bounded lattice;
2
(psBL ) ( , ,1)A is a monoid;
3
(psBL ) abc a b c b ac for any a,b,c A;
4
(psBL ) a b (ab)aa (a b);
5
(psBL ) (a b) (b a)(ab)(ba) 1 for any a,b A .
Operations , , operations have priority higher than , .
Let ( , , , , , 0,1)A a pseudo MV-algebra and let ,? two implications defined by:
x y y x (1)
xyxy (2)
then ( , , ,A ,, 0,1) is a pseudo BL-algebra. For every pseudo BL-algebra we denote
( ) { : }
G A x A xxx (3)
( ) { : ( ) ( ) }
M A x A x x x (4)
Be a Boolean algebra which contains the basic components of distributive lattice
( ) ( , , , 0,1)
L A A of pseudo BL-algebra A . so B A( )B L A( ( ))
Theorem 2Let ( )A set of ideals and a pseudo BL algebra A then we have the following equivalent statements:
(i) ( ( ), , , ,{1}, }A A Boolean algebra is;
(ii) Every ideal of A is principal for each a A there is n 1 so that its
a nB A( ).
Proof.
Fiabilitate si Durabilitate - Fiability & Durability Supplement no 1/ 2013 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 441
I know
(f 1/ f 1) ... (f m/ f )m 0. (5) We denote by
f=f 1 ... f m (6)
it follows that
a=f mI (7)
We denote by
f / =f 1/ ... f m
/
(8)
it follows that
a=f /mI. (9)
For i [1.. ] m we f f i si f / f i /
hence
f /
f
f i f i /
(10)
where 1 i m. therefore
(f /f)m f 1/ f 1) ... (f /m f m) (11) ie
(f /f)m0 (12)
it follows hat (f /
)m
f m0.
(13)
Shows that for a,b I we have that
b a 0 b a 0 (14) Let x I si b I then x b 1 .
But
a=1 a (a b) a (x a) (b a) x a (15) because b a 0.
So
a=x a a x x ( ]a (16)
as x I it follows that I ( ].a But a I ( ]a I .
therefore (a]=I , as I A and I ideal every ideal is principal in A.
For a A and A Boole algebra it follows that
(a] (a] A (a] ( ) a A (17)
{xA | x 1
1
( n ) ...( nm ),
m
a f a f
m 1, ,...,n1 nm0 , f1,...,fm (a] } A (18) but
1
1
( n ) ...( nm ) 0
m
a f a f (19)
Fiabilitate si Durabilitate - Fiability & Durability Supplement no 1/ 2013 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X 442
But
a fi a fi (20) ,for 1 i m.
Therefore a 1 ...
1 1
( ... ) ( ... )
m
n n
m m
f f f f
a n1 ... nm. (21)
but note n= a n1 ... nm and
1
(f ... fm) (a] (22)
therefore
0 0 (
n n n
a f f a f a an) (23) it follows that
f a ( an)a so an( an) 1 (24)
which implies that anB A( ).
" " show that ( ) A Boole is algebra.
I ( )A ,I {1} . How do I show that the principal ideal I= (a] .
As anB A( ) it follows that
( an) a 1 ( an) (a]={1} ( ( an) ) 1 0. (25)
But an ( a ) and ( a ) so an 0 . As 0 I it follows that I=A. Remark 3
I D As( )
We denote by
D D0D{1} { x A : ( ]x D {1}} (26) and a A,
(a] {x A : ( ]x ( ] {1}}.a (27)
If A is pseudo BL-algebra
D {x A : y x 1, for any y D} (28) and
(a] {x A : a x 1}. (29)
CONCLUSIONS
The problem addressed is the current approach as fuzzy systems underlying artificial intelligence is implemented in the economic and industrial machines.
REFERENCES
[1] Balbes si Dwinger,The curatours of the University of Missouri,1974
[2] Dumitru Buşneag ,Categories of Algebric Logic, Editura Academiei Române, Bucureşti
2006
[3] George Gratzer, Lattice teory,1971