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On some classes of BCI-algebras. (English) Zbl 0541.03037
In Proc. Japan Acad. 42, 26-29 (1966; Zbl 0207.293) K. Iséki introduced the concept of BCI-algebra. A BCI-algebra is an algebra $$ of type (2,0) satisfying the following conditions: (1) $\left(\left(x*y\right)*\left(x*z\right)\right)*\left(y*z\right)=0,$ (2) $\left(x*\left(x*y\right)\right)*y=0,$ (3) $x*x=0,$ (4) $x*y=y*x=0⇒x=y,$ (5) $x*0=0⇒x=0·$ In Math. Semin. Notes, Kobe Univ. 8, 125-130 (1980; Zbl 0434.03049) K. Iséki proved the following identities to hold in any BCI-algebra: (6) $\left(x*y\right)*z=\left(x*z\right)*y,$ (7) $x*0=x·$ In Math. Semin. Notes, Kove Univ. 8, 225- 226 (1980) K. Iséki showed that there is a variety of BCI- algebras. This variety is defined by the conditions (1), (2), (3), (6), (7) and (8) $\left(x*\left(x*y\right)\right)*\left(x*y\right)=y*\left(y*x\right)·$ These BCI-algebras are quasi- commutative and of type (1,0;0,0). In Math. Semin. Notes, Kobe Univ. 8, 553-555 (1980; Zbl 0473.03059) the authors introduced associative BCI- algebras. An associative BCI-algebra is a BCI-algebra satisfying (9) $\left(x*y\right)*z=x*\left(y*z\right)·$ In this paper the authors get the following result: Any associative BCI-algebra satisfies condition (8). The authors also show that there is a BCI-algebra which does not satisfy (8) giving a counterexample. This algebra satisfies: (10) $\left(\left(x*\left(x*y\right)\right)*\left(x*y\right)\right)*\left(x*y\right)=y*\left(y*x\right),$ (11) $\left(x*\left(x*y\right)\right)*\left(x*y\right)=\left(y*\left(y*x\right)\right)*\left(y*x\right)·$ It follows that there are quasi- commutative BCI-algebras of type (2,0;0,0) and (1,0;1,0). Thus the authors positively answer K. Isékis following question in Math. Semin. Notes, Kobe Univ. 8, 181-186 (1980; Zbl 0435.03043): Are there any quasi-commutative BCI-algebras of high type? Furthermore, the example shows that the class of quasi-commutative BCI-algebras of type (1,0;0,0) is a proper subclass of the class of all BCI-algebras. Thus, the variety above in K. Isékis sense is not the class of all BCI-algebras. Moreover, the BCI-algebra in the counterexample is not associative. Thus, this example shows that there is a new class of BCI-algebras.
##### MSC:
 03G25 Other algebras related to logic 06F99 Ordered structures (connections with other sections) 08A99 Universal algebra 08B99 Varieties of algebras 17D99 Other nonassociative rings and algebras