BCI-algebra

Algebras introduced by K. Iséki in [a4] as a generalized version of BCK-algebras (cf. BCK-algebra). The latter were developed by Iséki and S. Tannaka in [a6] to generalize the set difference in set theory, and by Y. Imai and Iséki in [a3] as the algebras of certain propositional calculi. It turns out that Abelian groups (cf. Abelian group) are a special case of BCI-algebras. One may take different axiom systems for BCI-algebras, and one such system says that a BCI-algebra is a non-empty set $X$ with a binary relation $\ast$ and a constant $0$ satisfying

i) ;

ii) ;

iii) $x \ast x = 0$;

iv) $x \ast y = 0$ and $y \ast x = 0$ imply that $x = y$;

v) $x \ast 0 = 0$ implies that $x=0$.

A partial order $\leq$ may be defined by $x \leq y$ if and only if $x \ast y = 0$. A very useful identity satisfied by $X$ is . One can then develop many of the usual algebraic concepts. An ideal is a set with the properties that and that whenever and , then . The ideal is implicative if and imply that . It is known that one always has . An ideal is closed if whenever then . While ideals in general are not subalgebras, closed ideals are. A subalgebra simply means a subset containing and closed under that is itself a BCI-algebra under .

The subset of all elements forms an ideal, called the -radical of . The algebra is a BCK-algebra if and only if , and is -semi-simple if and only if . In the latter case, satisfies the identity for all and . It then follows that one can define an operation on by , and . This makes into an Abelian group with as the identity. Conversely, every Abelian group can be given a BCI-algebra structure by . It follows that the category of Abelian groups is equivalent to the subcategory of the category of BCI-algebras formed by the -semi-simple BCI-algebras. Here, a homomorphism from one BCI-algebra to another is a function satisfying . In general, always contains a -semi-simple BCI-subalgebra, namely its -semi-simple part . Of course, also , since it can be verified easily that the induced partial order in a -semi-simple BCI-algebra is always trivial. Clearly, is -semi-simple if , and is a BCK-algebra if . Note that for a -semi-simple BCI-algebra, the closed ideals are precisely the subgroups of the associated Abelian group structure.

Some of the homological algebra properties of BCI-algebras are known. For example, it is known that a BCI-algebra is injective if and only if it is $p$-semi-simple and its associated Abelian group structure is divisible (cf, also Divisible group).

Fuzzy ideals of BCI-algebras are described in [a1] and [a2].

References

[a1]	C.S. Hoo, "Fuzzy ideals of BCI and MV-algebras" Fuzzy Sets and Systems , 62 (1994) pp. 111–114
[a2]	C.S. Hoo, "Fuzzy implicative and Boolean ideals of MV-algebras" Fuzzy Sets and Systems , 66 (1994) pp. 315–327
[a3]	Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 19–22
[a4]	K. Iséki, "An algebra related with a propositional calculus" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 26–29
[a5]	K. Iséki, "On BCI-algebras" Math. Seminar Notes (Kobe University) , 8 (1980) pp. 125–130
[a6]	K. Iséki, S. Tanaka, "An introduction to the theory of BCK-algebras" Math. Japon. , 23 (1978) pp. 1–26