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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
 
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b1101804.png" />;
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i) $\{ (x \ast y) \ast (x \ast z)\} \ast (z \ast y) = 0$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b1101805.png" />;
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ii) $\{ x \ast (x \ast y)\} \ast y = 0$;
  
 
iii) $x \ast x = 0$;
 
iii) $x \ast x = 0$;
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v) $x \ast 0 = 0$ implies that $x=0$.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018016.png" />. One can then develop many of the usual algebraic concepts. An [[Ideal|ideal]] is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018017.png" /> with the properties that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018018.png" /> and that whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018021.png" />. The ideal is implicative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018023.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018024.png" />. It is known that one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018025.png" />. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018026.png" /> is closed if whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018027.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018028.png" />. While ideals in general are not subalgebras, closed ideals are. A subalgebra simply means a subset containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018029.png" /> and closed under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018030.png" /> that is itself a BCI-algebra under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018031.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018032.png" /> of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018033.png" /> forms an ideal, called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018035.png" />-radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018036.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018037.png" /> is a [[BCK-algebra|BCK-algebra]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018039.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018041.png" />-semi-simple if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018042.png" />. In the latter case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018043.png" /> satisfies the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018044.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018046.png" />. It then follows that one can define an operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018048.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018050.png" />. This makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018051.png" /> into an [[Abelian group|Abelian group]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018052.png" /> as the identity. Conversely, every Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018053.png" /> can be given a BCI-algebra structure by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018054.png" />. It follows that the category of Abelian groups is equivalent to the subcategory of the category of BCI-algebras formed by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018055.png" />-semi-simple BCI-algebras. Here, a [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018056.png" /> from one BCI-algebra to another is a function satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018057.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018058.png" /> always contains a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018059.png" />-semi-simple BCI-subalgebra, namely its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018060.png" />-semi-simple part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018061.png" />. Of course, also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018062.png" />, since it can be verified easily that the induced partial order in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018063.png" />-semi-simple BCI-algebra is always trivial. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018064.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018065.png" />-semi-simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018066.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018067.png" /> is a BCK-algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018068.png" />. Note that for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018069.png" />-semi-simple BCI-algebra, the closed ideals are precisely the subgroups of the associated Abelian group structure.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110180/b11018070.png" />-semi-simple and its associated Abelian group structure is divisible (cf, also [[Divisible group|Divisible group]]).
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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]]].
 
Fuzzy ideals of BCI-algebras are described in [[#References|[a1]]] and [[#References|[a2]]].
  
 
====References====
 
====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</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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<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>
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<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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Latest revision as of 02:33, 15 February 2024

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
How to Cite This Entry:
BCI-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BCI-algebra&oldid=54782
This article was adapted from an original article by C.S. Hoo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article