Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Citations: Zbl 0435.03044; Zbl 0207.293; Zbl 0434.03049; Zbl 0473.03059; Zbl 0435.03043

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]