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Zbl 0541.03037
Hu, Qing Ping; Iséki, Kiyoshi
(Hu, Quingping)
On some classes of BCI-algebras.
(English)
[J] Math. Jap. 29, 251-253 (1984). ISSN 0025-5513

In Proc. Japan Acad. 42, 26-29 (1966; Zbl 0207.293) {\it K. Iséki} introduced the concept of BCI-algebra. A BCI-algebra is an algebra $<X;*,0>$ of type (2,0) satisfying the following conditions: (1) $((x*y)*(x*z))*(y*z)=0,$ (2) $(x*(x*y))*y=0,$ (3) $x*x=0,$ (4) $x*y=y*x=0\Rightarrow x=y,$ (5) $x*0=0\Rightarrow x=0.$ In Math. Semin. Notes, Kobe Univ. 8, 125-130 (1980; Zbl 0434.03049) {\it K. Iséki} proved the following identities to hold in any BCI-algebra: (6) $(x*y)*z=(x*z)*y,$ (7) $x*0=x.$ In Math. Semin. Notes, Kove Univ. 8, 225- 226 (1980) {\it 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) $(x*(x*y))*(x*y)=y*(y*x).$ 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) $(x*y)*z=x*(y*z).$ 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) $((x*(x*y))*(x*y))*(x*y)=y*(y*x),$ (11) $(x*(x*y))*(x*y)=(y*(y*x))*(y*x).$ 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 {\it 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 2000:
*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

Keywords: BCI-algebra; variety; associative BCI-algebra; quasi-commutative BCI- algebras

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