# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
BCI-algebras and Abelian groups. (English) Zbl 0636.03059

If the BCK-part of a BCI-algebra (X,$·,0\right)$ is trivial, then $\left(X,+\right)$, where $x+y=x\left(Oy\right)$, is an abelian group. Conversely, if $\left(X,+\right)$ is an abelian group, then (X,$·,O\right)$, where $xy=y-x$, is a BCI-algebra and its BCK-part is trivial.

Reviewer’s remark: This result was presented by the reviewer during the All-Polish Conference on Universal Algebra and its Applications, Opole, May 1985 (see Materiały or Demonstr. Math. (appear)).

Reviewer: W.A.Dudek
##### MSC:
 03G25 Other algebras related to logic 20L05 Groupoids 08A05 Structure theory of general algebraic systems
##### Keywords:
BCK-part; BCI-algebra; abelian group