BCK algebra

In mathematics, BCI and BCK algebras are algebraic structures, introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics.

Definition

BCI algebra

An algebra $left(\; X;ast,0\; ight)$ of type $left(\; 2,0\; ight)$ is called a "BCI-algebra" if, for any $x,y,zin\; X$, it satisfies the following conditions:; BCI-1: $left(\; left(\; xast\; y\; ight)\; ast\; left(\; xast\; z\; ight)\; ight)\; ast\; left(\; zast\; y\; ight)\; =0$; BCI-2: $left(\; xast\; left(\; xast\; y\; ight)\; ight)\; ast\; y=0$; BCI-3: $xast\; x=0$; BCI-4: $xast\; y=0\; and\; yast\; x=0implies\; x=y$; BCI-5: $xast\; 0=0\; implies\; x=0$

BCK algebra

A BCI-algebra $left(\; X;ast\; ,0\; ight)$ is called a "BCK-algebra" if itsatisfies the following condition:; BCK-1: $forall\; xin\; X:\; 0ast\; x=0$

Examples

Every abelian group of is a BCI-algebra, with * group subtraction and 0 the group identity.

The subsets of a set form a BCK-algebra, where A*B is the difference AB (elements in A but not in B), and 0 is the empty set.

A Boolean algebra is a BCK algebra if "A"*"B" is defined to be "A"∧¬"B" ("A" does not imply "B").

References

*citation|title=Review of several papers on BCI, BCK-Algebras
first= R. B. |last=Angell
journal= The Journal of Symbolic Logic|volume= 35|issue= 3|year=1970|pages= 465-466
url= http://links.jstor.org/sici?sici=0022-4812%28197009%2935%3A3%3C465%3AAARWAP%3E2.0.CO%3B2-C
*citation|url=http://projecteuclid.org/euclid.pja/1195522126|id=MR|0202572
last=Arai|first= Yoshinari|last2= Iséki|first2= Kiyoshi|last3= Tanaka|first3= Shôtarô
title=Characterizations of BCI, BCK-algebras
journal=Proc. Japan Acad. |volume=42|year= 1966 |pages=105-107
*springer|id=B/b110170|title=BCH algebra|first=C.S.|last= Hoo
*springer|id=B/b110180|title=BCI algebra|first=C.S.|last= Hoo
*springer|id=B/b110190|first=C.S.|last= Hoo
*citation|first=K. |last=Iséki|first2= S.|last2= Tanaka|title=An introduction to the theory of BCK-algebras" |journal=Math. Japon. |volume= 23 |year=1978|pages= 1–26
* Y. Huang, "BCI-algebra", Science Press, Beijing, 2006.
*citation|first=Y.|last= Imai|first2= K|last2= Iséki|title=On axiom systems of propositional calculi, XIV |journal=Proc. Japan Acad. Ser. A, Math. Sci. |volume= 42 |year=1966|pages= 19–22 |url=http://projecteuclid.org/euclid.pja/1195522169
*citation|first=K. |last=Iséki|title=An algebra related with a propositional calculus|journal= Proc. Japan Acad. Ser. A, Math. Sci. |volume= 42 |year=1966|pages= 26–29|url=http://projecteuclid.org/euclid.pja/1195522171