View source for BCI-algebra


Algebras introduced by K. Iséki in [[#References|[a4]]] as a generalized version of BCK-algebras (cf. [[BCK-algebra|BCK-algebra]]). The latter were developed by Iséki and S. Tannaka in [[#References|[a6]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[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|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|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|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|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 [[#References|[a1]]] and [[#References|[a2]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.S. Hoo,   "Fuzzy ideals of BCI and MV-algebras"  ''Fuzzy Sets and Systems'' , '''62'''  (1994)  pp. 111–114</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.S. Hoo,   "Fuzzy implicative and Boolean ideals of MV-algebras"  ''Fuzzy Sets and Systems'' , '''66'''  (1994)  pp. 315–327</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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}}</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Iséki,   "An algebra related with a propositional calculus"  ''Proc. Japan Acad. Ser. A, Math. Sci.'' , '''42'''  (1966)  pp. 26–29</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Iséki,   "On BCI-algebras"  ''Math. Seminar Notes (Kobe University)'' , '''8'''  (1980)  pp. 125–130</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Iséki,   S. Tanaka,   "An introduction to the theory of BCK-algebras"  ''Math. Japon.'' , '''23'''  (1978)  pp. 1–26</TD></TR>
</table>

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