ON GROUP-LIKE BCI-ALGEBRAS

1, Bj a Β CI -algebra iva mean a non-ecipty set G together ν,i'ii a binary multiplicación denoted by juxtaposition and 'some distinguished element 0 such that the following axioms are satisf ied: 1; ( <>.y ) (xz) ) í7y j ~ 0, (2) xy = yx O implies χ = y, ί3) xO χ. This axiome sy¿;te.n is inöepenoent (c f . [ 3 ] ) · As a simple ccnsec)uencs of the afccvc í j c í o u l í system (c f . [ 3 ] » [ 6 J ? we obtains (4) (x(xy) )y = Of (5) XX = 0, (6) (xy)z = ( x*)y. A BCI-algebra is calleó a BC'*.-alga bra i f i t sat is f ies the identity (7) Ox = 0. Tha cles3 zi a l l BCK-sigabras is a prbper subquaeivariety of the quasivariely o? a l l K l -a lgebras . This sub^uasivarie".y is uniquely detìTì-.nac oy "-he indeoencent axioms system: (1 ) , (2 ) , (3) ami ("') (o f . [3] h On the ether .'itrvî, o.ie e n ¿rove täat the quasivarioty of a l l BCI-algeüras coniai v. Bone cxasjes cf BCI-algebras which forti t> vaviîîy F ·ν «.Brit? t¡ae class cf a l l medial BCI -ai", e »rat' cìj. ¡2j ¡ , i . e . JCI-alge brae with the additional identity