zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 <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=0x=y, (5) x*0=0x=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) (x*y)*z=(x*z)*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) (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 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.
03G25Other algebras related to logic
06F99Ordered structures (connections with other sections)
08A99Universal algebra
08B99Varieties of algebras
17D99Other nonassociative rings and algebras