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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 ) abca   b c b ac for any a,b,cA;

4

(psBL ) a b (ab)aa (a b);

5

(psBL ) (a  b) (b a)(ab)(ba) 1 for any a,bA .

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)

xyxy (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 xx   x  (4)

Be a Boolean algebra which contains the basic components of distributive lattice

( ) ( , , , 0,1)

L AA  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.

(2)

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)m0 (12)

it follows hat (f /

)m

 f m0.

(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 ( ]aI .

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 nm0 , f1,...,fm (a] } A (18) but

1

1

( n ) ...( nm ) 0

m

afaf  (19)

(3)

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

affa   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 D0D{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

Referências

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