Walendziak, Andrzej

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaAndrzej WalendziakSome axiomatizations of B-algebrasMathematica Slovaca, Vol. 56 (2006), No. 3, 301--306Persistent URL: http://dml.cz/dmlcz/131319Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2006Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca © 2006 hň -.L <ri r £ /~r\r\<z\ M O O/-\I i/-..- Mathematical Insti tute Math . SlOVaCa, 5 6 ( 2006 ) , NO. 3 , 3 0 1 - 3 0 6 Slovák Academy of Sciences SOME AXIOMATIZATIONS OF B-ALGEBRAS A N D R Z E J WALENDZIAK (Communicated by Anatolij Dvurecenskij ) ABSTRACT. Some systems of axioms defining a B-algebra are given with a proof of the independence of the axioms. In addition, we obtain a simplified axiomatization of commutative B-algebras. 1. Introduction P?-algebras have been introduced by J. N e g g e r s and H. S. K im in [4]. They defined a B-algebra as an algebra (A, *,0) of type (2,0) satisfying the following axioms: Al. x * x = 0, A2. x *0 = x, A3, (x * y) * z = x * (z * (0 * y)). We will denote by B the class of all B-algebras. In [1], J. R. Cho and H. S. K im proved that every J3-algebra is a quasigroup. M. K o n d o and Y. B. J u n [3] showed that the class B is equivalent in one sense to the class of groups. In [2], Y. B. J u n, E. H. R o h and H. S. K i m introduced the notion of BJf-algebras, which is a generalization of BCH jBCI/BCK'-algebras. More-over, B is a proper subclass of the class of BiJ-algebras (cf. [4; Lemma 2.9]). For another useful generalization of B -algebras see [6]. 2. Some axiomatizations of B -algebras THEOREM 2.1. Let (A, - , + , 0 ) be an algebra of type (2,2,0) satisfying the following axioms: Bl. x-x = 0, B2. x-0 = x, B3. (x-y)-z = x-(z + y), B4. x + y = x-(0-y). Then (A, —,0) is a B-algebra. 2000 M a t h e m a t i c s Sub jec t C l a s s i f i c a t i o n : Primary 06F35. Keywords : B-algebra, commutative B-algebra. 301 ANDRZEJ WALENDZIAK Conversely, if (A, — ,0) G B. and if we define x + y by x * (0 * y), then (A, —, +, 0) obeys the equations B1-B4. P r o o f . Straightforward. • In [6], J. N e g g e r s and H. S. K i m introduced the notion of /3-algebras. They defined a f3-algebra as an algebra (A, —, +, 0) of type (2, 2, 0) that obeys B2, B3, and the following axiom: (0-x) + x = 0. It is easy to verify that if an algebra (A, —, +, 0) satisfies B1-B4, then it is a (3 -algebra. THEOREM 2.2. Let A = (A, *,0) be an algebra of type (2,0) . Then A e B if and only if A obeys the laws: CI. x * x = 0. C2. 0 * ( 0 * x ) = x, C3. (x * z) * (y * z) = x * y. P r o o f . Suppose that A is a 5-algebra. For each x e A we have 0 * (0 * x) = x (see [4; Lemma 2.9]). Consequently, C2 is valid in A . By A3 we obtain (x * z) * (y * z) = x * [(y * z) * (0 * z)] = x * [H * ((0 * z) * (0 * z))] . Hence applying Al and A2 we get C3. Conversely, assume that C1-C3 hold in A . Then we have x = 0 * (0 * x) = (x * x) * (0 * x) = x * 0 . From this and from C3 we deduce that (x * y) * (0 * y) = x . (1) Combining (1) with C3 we get x * (z * (0 * y)) = [(x * y) * (0 * y)] * [z * (0 * y)] = (x * y) * z, i.e., A3 holds. Therefore A e B. • LEMMA 2.3. Let (A, *,0) 6e an algebra of type (2,0) obeying the following laws: Dl . x * x = 0. D2. x * J [(0 * y) * z] * [(0 * x) * z] \ = 2/ • (i) x * 0 = x, (ii) 0 * (0 * x) = x, (iii) 0 * x = 0 * y => x = y. (iv) (x * 2/) * (0 * y) = x, (v) x * 2/ = 0 * (y * x ) . 302 SOME AXIOMATIZATIONS OF B-ALGEBRAS P r o o f . (i): To obtain (i), substitute x for y in D2 and then use Dl. (ii): Substituting x = 0, y = x, and z = 0, D2 becomes 0 * { [(0 * x) * 0] * [(0 * 0) * 0] } = x . Applying (i) we obtain (ii). (iii) follows from (ii). (iv): Let a, b G A. Using D2 with x = 0, H = 0 * a, z = b we have 0 * { [(0 * (0 * a)) * b] * [(0 * 0) * b] } = 0 * a. Hence applying (i) and (ii) we conclude that 0 * [(a * b) * (0 * b)] = 0 * a . That (a * b) * (0 * b) = a follows from (iii). (v): Let a, b G A. Substituting x = a, y = 0 * (b * a), z = 0 in D2 we deduce that a * { [(0 * (0 * (b * a))) * 0] * [(0 * a) * 0] } = 0 * (b * a) . Then a* [(b*a) * (0*a)] = 0* (b*a). By (iv), a*b = 0*(b*a), verifying (v). a THEOREM 2.4. An algebra A = (A, *,0) of type (2,0) is a B-algebra if and only if the equations Dl and D2 are valid in A. P r o o f . Let A satisfy Dl and D2. CI holds in A by Dl. From Lemma 2.3 (iii) we conclude that A obeys C2. If we let x = a*c, y = a*b and z = 0 * a in D2, then we have (a* c) * I [(0 * (a * b)) * (0 * a)] * [(0 * (a* c)) * (0 * a)] } = a * b. By Lemma 2.3, (0 * (a * b)) * (0 * a) = (b * a) * (0 * a) = b, and similarly, (0 * (a * c)) * (0 * a) = c. Consequently, (a* c) * (b * c) = a * b. This shows that A also satisfies C3. Then A G B by Theorem 2.2. For the converse, suppose that A is a B-algebra. Obviously Dl is valid in A. From Theorem 2.2 we see that C3 holds in A, and therefore [(0 * y) * z] * [(0 * x) * z] = (0 * y) * (0 * x). (2) 303 ANDRZEJ WALENDZIAK It follows that x * { [(0 * y) * z] * [(0 * x) * z] | = x * [(0 * y) * (0 * x)] (by (2)) = (x * x) * (0 * y) (by A3) = 0 * (0 * y) (by Al) = V (byC2), proving D2. The proof is finished. • Following J. N e g g e r s and H. S. K i m [4] (see also [1]) wre give: DEFINITION 2.5. A B-algebra (A, *, 0) is said to be 0-commutative if a*(0*b) = b * (0 * a) for all a, b G A. In [1], J. R. C l io and H. S. K i m showed that a S-algebra A = (A, *,0) is 0-commutative if and only if the equation C2' y * (y * x) = x holds in A . From this and from Theorem 2.2 we have: COROLLARY 2.6. An algebra (A, *,0) of type (2,0) is a 0-commutative B -algebra if and only if it obeys the laws CI, C2 ' . and C3. 3. Proof of the independence of the axioms The independence of the axioms Al , A2, and A3 was proved by J. N e g g e r s and H. S. K i m in [4]. THEOREM 3 .1 . The axioms B1-B4 are independent, i.e., none of them can be deduced from the others. P r o o f . We are going to give some examples of algebras in which only three of the axioms hold. Let A — {0,1}. Define binary operations 0 and 0 on A as follows: x Qy = x for all x ,y G A, x 0 y = 0 for all x,y G A. Then (A, 0 , 0 , 0 ) fulfils the axioms B2-B4, but not Bl , since 1 0 1 = 1 ^ - 0 . (Independence of Bl.) It is easily seen that (A, 0 , 0 ,0 ) satisfies Bl , B3, and B4, but not B2 (inde-pendence of B2). 304 SOME AXIOMATIZATIONS OF B-ALGEBRAS Now we define the binary operations — and + on A by the following table. X У x-y x + y 0 0 0 0 0 1 0 0 1 0 1 1 1 1 0 1 The equations Bl , B2, and B4 are valid in (A, —, +, 0), but B3 does not hold because (1 - 1) - 0 = 0, while 1 - (0 + 1) = 1. (Independence of B3.) Finally, let A = (A, — , + , 0 ) be the algebra, where — is given in the above table and + is defined by 0 if x = y = 0 , otherwise. x + y = {l LI...!. " (3) Obviously, B1-B3 hold in A , while B4 does not (independence of B4). D THEOREM 3.2. The system of axioms C1-C3 is independent. P r o o f . Let A = {0,1}. We use the table below in order to define *, and * . X У x * y x ®y x * y 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 0 We can see that the algebra (A, *, 0) satisfies C2-C3, but not CI . The axioms CI and C3 hold in (A, ®, 0), while C2 does not. It is evident that (A, * , 0) obeys CI and C2. The axiom C3 does not hold because (0 * 1) * (1 * 1) = 0, while 0 * 1 = 1. D Remark 3.3. It is easy to see that the axiom system D1-D2 of B-algebras is independent. 305 ANDRZEJ WALENDZIAK R E F E R E N C E S [1] CHO, J. R.—KIM, H. S.: On B -algebras and quasigroups, Quasigroups Rela ted Systems 7 (2001), 1-6. [2] JUN, Y. B.—ROH, E. H.—KIM, H. S.: On BH-algebras, Sci. Ma th. Jpn. 1 (1998), 347-354. [3] K O N D O , M.—JUN, Y. B . : The class of B-algebras coincides with the class of groups, Sci. Ma th. Jpn. 5 7 (2003), 197-199. [4] NEGGERS, J .—KIM, H. S.: On B-algebras, Ma t . Vesnik 54 (2002), 21-29. [5] NEGGERS, J .—KIM, H. S. : A fundamental theorem of B -homomorphism for B -algebras, Int. Ma th. J. 2 (2002), 207-214. [6] NEGGERS, J .—KIM, H. S.: On (3-algebras, Ma th. Slovaca 52 (2002), 517-530. Received Augus t 23, 2004 Warsaw School of Information Technology Newelska 6 Revised Oc tober 8, 2004 PL-01-447 Warszawa POLAND University of Podlasie 3 Maja 54 PL-08110 Siedlce POLAND E-mail: walent@interia.pl 306 

Some axiomatizations of $B$-algebras

http://dml.cz/bitstream/handle/10338.dmlcz/131319/MathSlov_56-2006-3_5.pdf

Some axiomatizations of $B$ -algebras

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Some axiomatizations of BBB-algebras

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Some axiomatizations of $B$ -algebras