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) $\{ (x \ast y) \ast (x \ast z)\} \ast (z \ast y) = 0$;

ii) $\{ x \ast (x \ast y)\} \ast y = 0$;

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 $(x \ast y) \ast z = (x \ast z) \ast y$. One can then develop many of the usual algebraic concepts. An ideal is a set $I$ with the properties that $0 \in I$ and that whenever $x \ast y \in I$ and $y \in I$, then $x \in I$. The ideal is implicative if $(x \ast y) \ast z \in I$ and $y \ast z \in I$ imply that $x \ast z \in I$. It is known that one always has $(x \ast z) \ast z \in I$. An ideal $I$ is closed if whenever $x \in I$ then $0 \ast x \in I$. While ideals in general are not subalgebras, closed ideals are. A subalgebra simply means a subset containing $0$ and closed under $\ast$ that is itself a BCI-algebra under $\ast$.

The subset $X_+$ of all elements $x \ge 0$ forms an ideal, called the $p$-radical of $X$. The algebra $X$ is a BCK-algebra if and only if $X = X_+$, and $X$ is $p$-semi-simple if and only if $X_+ = \{0\}$. In the latter case, $X$ satisfies the identity $x \ast (0 \ast y) = y \ast (0 \ast x)$ for all $x$ and $y$. It then follows that one can define an operation $+$ on $X$ by $x + y = x \ast (0 \ast y)$, and $-x = 0 \ast x$. This makes $X$ into an Abelian group with $0$ as the identity. Conversely, every Abelian group $(X, +, 0)$ can be given a BCI-algebra structure by $x \ast y = x - y$. It follows that the category of Abelian groups is equivalent to the subcategory of the category of BCI-algebras formed by the $p$-semi-simple BCI-algebras. Here, a homomorphism $f : X \to Y$ from one BCI-algebra to another is a function satisfying $f(x\ast y) = f(x) \ast f(y)$. In general, $X$ always contains a $p$-semi-simple BCI-subalgebra, namely its $p$-semi-simple part $X_p = \{x \in X: 0 \ast (0 \ast x) = x\}$. Of course, also $X_p = \{x \in X : y \le x \implies y = x\}$, since it can be verified easily that the induced partial order in a $p$-semi-simple BCI-algebra is always trivial. Clearly, $X$ is $p$-semi-simple if $X = X_p$, and $X$ is a BCK-algebra if $X_p = \{0\}$. Note that for a $p$-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 Zbl 0156.24812
[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