PSEUDO MV-ALGEBRAS SI PSEUDO BL-ALGEBRAS

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Fiabilitate si Durabilitate - Fiability &_{Durability Supplement no 1/ 2013} Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X

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PSEUDO MV-ALGEBRAS SI PSEUDO BL-ALGEBRAS

Constantin BOGDAN

Department of Mathematics, University of Craiova, 200585 Craiova, Dolj, Romania EMAL tica1234ticabogd@yahoo.com

Asistent univ. Olimpia-Mioara PECINGINA Univ. C-Tin Brancusi Tg-Jiu

Abstract. _{This material is studying MV-algebras and BL-algebras in terms of applicability in Boole} algebra.Boole algebras expanded use in fuzzy logic.

Pseudo MV-algebras

We consider an algebra  ( , , , , 0,1)A   of type (2,1,1, 0, 0). We define:

( )

yx xy  consider that the operation  takes precedence over surgery  .

Definition 1 A pseudo MV-algebra is an algebra  ( ,A , , , 0 , 1  of type (2,1,1, 0, 0) satisfying the following axioms:

1

(psMV) x    (y z) (x y) z;

2

(psMV ) x   0 0 x x;

3

(psMV ) x   1 1 x 1;

4

(psMV ) 1 0;1 0;

5

(psMV ) (xy ) (xy ) ;

6

(psMV ) xxy y yxxy y yxx;

7

(psMV ) x(xy) (x y)y;

8

(psMV ) (x ) x , for all , ,x y zA .

We denote a pseudo MV-algebra  ( , , , , 0,1)A  the universe A . We define two implications corresponding to the two negations:

:

xy xy si xy : y x for all x y, A .

A pseudo MV-algebra is non-trivial provided that the universe have more than one item.

Remark 2

Let a,b B A( )

And a=ba b (1)

a b a

    a=b a a. (2)

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1=(e ee)(ee)(ee ) (3) 1=(e e)(e e )e(ee)(ee) (4)

Remark 4

In the same conditions as in Theorem 3 we have that:

( )

ee e e e (5)

( )

ee e e e (6)

Denote y= (ee)e (7)

ee   y e y e (8)

Dar ee y  e  y e (9) So (ee) y e (10)

Show that (ee)e y (11) Therefore

(ee)e (e e)(ee). (12) Proof of Theorem 3:

Applying the four outstanding results and building on gender ee 1 , the left term is equal

( ee)( ee) 1. (13)

The term on the right of equality in Theorem 3 becomes

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

e e e e e e e e e e

e e e e e e

  

  

  

   

     

   (14)

(ee)(ee)(ee ) . (15)

Theorem 5 Let A be a pseudo MV algebra. If e B A( ) and e  e e then e(ee)(ee) (e e) (16)

Remark 6 Show that the same conditions as the previous theorem we have equality

e  (e e) e e (17)

or

e  (e e) e e (18)

On the other hand, if we denote by y=e  (e e) we have that y ( )e e (19)

and

y ( )e e (20)

 ( )e  y e and e y e (21) so e y  e e. (22)

We have shown that

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Returning to the notation

(e e)  e e e (24)

And

(e e)  e e e. (25)

Proof of Theorem 5:

e  (e e) e [e(ee)] e (ee). (26) Evaluate the expression

e  (e e)e[(ee )  (e e)] (27)

( ) ( )

e e e e e

 _ _    (28)

(ee) (e e)[(ee ) e)] (e e) (29) ( ?e e) (e e).

   (30)

Theorem 7 Let A be a pseudo MV algebra. If e B A( ) si e ?ee then

( )

e ee ee (31)

Proposition 8 We have the following expressions true:

( )

eee ee (32)

( )

ee e ee (33)

Denote

y= e(ee) (34)

ee  y eey (35)

ee y eey (36)

So eeey. (37) Show that

( )

e ee y (38)

Get

( ) ( ) ( ).

e ee  ee  ee (39) The proof of Theorem 7:

( )

e ee  e ( e (ee)) (40) (e e) (e e)

 _{ }   e(ee). (41) Processed expression

( ) ( ) ( )

e ee  e e  e e (42) (e e) (e e)

     (43)

( ee) (e e)( )e  (e e) (44) (e e) e e e 0.

      (45)

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Definition 9 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 ) abc  a   b c b ac for all a,b,c  A;

4

(psBL ) a b (ab)aa (a b);

5

(psBL ) (a  b) (b a)(ab)(ba) 1 for all 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 (46)

xyxy (47) Then

( , , ,A   ,, 0,1) is a pseudo BL-algebra. For every pseudo BL-algebra A denote

( ) { : }

G A  x A xxx (48)

( ) { : ( ) ( ) }

M A  x A x x   x  (49)

Let B A( ) 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 10 Let A be an algebra and Pseudo BL a,b  A we have the following statements true:

(i) (a b )  a b (50) (ii) (a b ) ba (51) Proof.

(i) Show that

(a b) 0 ( a b)0 (52) (a b)0 a b 0 (53) (a b)0] a b0 (54) (a b) 0 a b 0. (55)

(ii) Show that

(a  b) 0b (a 0) (56)

b [ (a  b) 0](a0) (57)

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Theorem 11 Fie A o Pseudo BL algebra .Then we have the equivalent:

e B A( )  e (ee)ee (60)

Theorem 12

Let A be a pseudo BL algebra. Then we have equivalence:

e B A( )  e (ee ) ee (61)

Remark 13 Taking a = b = e in Theorem 10, (i) obtain proof of the theorem 11.

Remark 14 Taking a = b = e in Theorem 10, (ii) we obtain the proof of Theorem 12.

CONCLUSIONS

The material has great resonance in fuzzy systems used in robotics because there is a pseudo BL-algebras expansion and pseudo MV-algebras

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