Namespaces
Variants
Actions

Difference between revisions of "BCK-algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex done)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
Algebras originally defined by K. Iséki and S. Tanaka in [[#References|[a7]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[a5]]] as the algebras of certain propositional calculi. A BCK-algebra may be defined as a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101901.png" /> with a [[Binary relation|binary relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101902.png" /> and a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101903.png" /> satisfying the following axioms:
 
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101904.png" />;
+
Algebras originally defined by K. Iséki and S. Tanaka in [[#References|[a7]]] to generalize the set difference in set theory, and by Y. Imai and Iséki in [[#References|[a5]]] as the algebras of certain propositional calculi. A BCK-algebra may be defined as a non-empty set $X$ with a [[binary relation]] $\ast$ and a constant $0$ satisfying the following axioms:
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101905.png" />;
+
1) $\{(x\ast y) \ast (x\ast z)\} \ast (z \ast y) = 0$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101906.png" />;
+
2) $\{x \ast (x \ast y) \} \ast y = 0$;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101908.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b1101909.png" />;
+
3) $x \ast x = 0$;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019011.png" />;
+
4) $x \ast y = 0$ and $y \ast x = 0$ imply $x = y$;
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019013.png" />. A [[Partial order|partial order]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019014.png" /> can then be defined by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019015.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019016.png" />. A very useful property is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019017.png" />.
+
5) $x \ast 0 = 0$ implies $x = 0$;
  
A BCK-algebra is commutative if it satisfies the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019018.png" /> (cf. also [[Commutative ring|Commutative ring]]). In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019019.png" />, the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019021.png" /> under the partial order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019022.png" />. The BCK-algebra is bounded if it has a largest element. Denoting this element by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019023.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019024.png" />, the least upper bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019026.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019027.png" /> is a [[Distributive lattice|distributive lattice]] with bounds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019029.png" />. A BCK-algebra is positive implicative if it satisfies the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019030.png" />. This is equivalent to the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019031.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019032.png" /> is called implicative if it satisfies the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019033.png" />. Every implicative BCK-algebra is commutative and positive implicative, and a bounded implicative BCK-algebra is a [[Boolean algebra|Boolean algebra]].
+
6) $0 \ast x = 0$ for all $x$. A [[Partial order|partial order]] $\le$ can then be defined by putting $x \le y$ if and only if $x \ast y = 0$. A very useful property is $(x \ast y) \ast z = (x \ast z) \ast y$.
  
An [[Ideal|ideal]] of a BCK-algebra is a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019035.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019037.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019038.png" />. The ideal is implicative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019040.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019041.png" />. It is known that always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019042.png" />. Note that in a positive implicative BCK-algebra, every ideal is implicative. Implicative ideals are important because in a bounded commutative BCK-algebra they are precisely the ideals for which the quotient BCK-algebras are Boolean algebras. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019043.png" /> is an ideal in a BCK-algebra, one can define a [[Congruence|congruence]] relation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019044.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019045.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019047.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019048.png" /> of congruence classes then becomes a BCK-algebra under the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019049.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019050.png" /> as the constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019051.png" /> as the largest element if there exists a largest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019052.png" />. Some, but not all, of the well-known results on distributive lattices and Boolean algebras hold in BCK-algebras, in particular in bounded commutative BCK-algebras. For example, the prime ideal theorem holds for bounded commutative BCK-algebras, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019053.png" /> is an ideal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019054.png" /> is a lattice [[Filter|filter]] such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019055.png" />, then there exists a [[Prime ideal|prime ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019056.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019058.png" />. Here,  "prime ideal"  simply means that if it contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019059.png" />, then it contains either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019060.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019061.png" />.
+
A BCK-algebra is commutative if it satisfies the identity $x \ast (x \ast y) = y \ast (y\ast x)$ (cf. also [[Commutative ring|Commutative ring]]). In this case, $x \ast (x \ast y) = x \wedge y$, the greatest lower bound of $x$ and $y$ under the partial order $\le$. The BCK-algebra is bounded if it has a largest element. Denoting this element by $1$, one has $1 \ast \{(1\ast x) \wedge (1\ast y)\} = x \vee y$, the least upper bound of $x$ and $y$. In this case, $X$ is a [[Distributive lattice|distributive lattice]] with bounds $0$ and $1$. A BCK-algebra is positive implicative if it satisfies the identity $(x\ast y)\ast z = (x\ast z) \ast (y\ast z)$. This is equivalent to the identity $x \ast y = (x\ast y)\ast y$. $X$ is called implicative if it satisfies the identity $x \ast (y \ast x) = x$. Every implicative BCK-algebra is commutative and positive implicative, and a bounded implicative BCK-algebra is a [[Boolean algebra|Boolean algebra]].
  
Some of the homological algebra properties of BCK-algebras are known, see [[#References|[a2]]]. There is also a close connection between BCK-algebras and commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019062.png" />-groups with order units (cf. [[L-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019063.png" />-group]]). Recall that an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019064.png" /> in the positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019065.png" /> of a commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019066.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019067.png" /> is an order unit if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019068.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019069.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019070.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019071.png" /> For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019073.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110190/b11019074.png" /> is a commutative BCK-algebra.
+
An [[Ideal|ideal]] of a BCK-algebra is a non-empty set $I$ such that $0 \in I$ and if $x \ast y \in I$ and $y \in I$ imply $x \in I$. The ideal is implicative if $(x\ast y) \ast z \in I$ and $y \ast z \in I$ imply $x \ast z \in I$. It is known that always $(x \ast z) \ast z \in I$. Note that in a positive implicative BCK-algebra, every ideal is implicative. Implicative ideals are important because in a bounded commutative BCK-algebra they are precisely the ideals for which the quotient BCK-algebras are Boolean algebras. Here, if $I$ is an ideal in a BCK-algebra, one can define a [[Congruence|congruence]] relation in $X$ by $x \sim y$ if and only if $x \ast y \in I$ and $y \ast x \in I$. The set $X/I$ of congruence classes then becomes a BCK-algebra under the operation $[x]\ast [y] = [x\ast y]$, with $[0]$ as the constant and $[1]$ as the largest element if there exists a largest element $1$. Some, but not all, of the well-known results on distributive lattices and Boolean algebras hold in BCK-algebras, in particular in bounded commutative BCK-algebras. For example, the prime ideal theorem holds for bounded commutative BCK-algebras, that is, if $I$ is an ideal and $F$ is a lattice [[Filter|filter]] such that $I \cap F = \emptyset$, then there exists a [[Prime ideal|prime ideal]] $J$ such that $I \subset J$ and $J \cap F = \emptyset$. Here,  "prime ideal"  simply means that if it contains $x \wedge y$, then it contains either $x$ or $y$.
 +
 
 +
Some of the homological algebra properties of BCK-algebras are known, see [[#References|[a2]]]. There is also a close connection between BCK-algebras and commutative $l$-groups with order units (cf. [[L-group|$l$-group]]). Recall that an element $u$ in the positive cone $G_+$ of a commutative $l$-group $G$ is an order unit if for each $x \in G$ one has $x \le nu$ for some integer $n$. Let $G(u) = \{x \in G : 0 \le x < u\}$.  
 +
For $x, y \in G(u)$, let $x \ast y = (x-y)_+ = (x-y) \vee 0$. Then $G(u)$ is a commutative BCK-algebra.
  
 
Fuzzy ideals of BCK-algebras are described in [[#References|[a3]]] and [[#References|[a4]]]. General references for BCK-algebras are [[#References|[a6]]] and [[#References|[a7]]].
 
Fuzzy ideals of BCK-algebras are described in [[#References|[a3]]] and [[#References|[a4]]]. General references for BCK-algebras are [[#References|[a6]]] and [[#References|[a7]]].
Line 23: Line 25:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.S. Hoo,  P.V. Ramana Murty,  "The ideals of a bounded commutative BCK-algebra"  ''Math. Japon.'' , '''32'''  (1987)  pp. 723–733</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.S. Hoo,  "Injectives in the categories of BCK and BCI-algebras"  ''Math. Japon.'' , '''33'''  (1988)  pp. 237–246</TD></TR><TR><TD valign="top">[a3]</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">[a4]</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">[a5]</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">[a6]</TD> <TD valign="top">  K. Iséki,  S. Tanaka,  "Ideal theory of BCK-algebras"  ''Math. Japon.'' , '''21'''  (1976)  pp. 351–366</TD></TR><TR><TD valign="top">[a7]</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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.S. Hoo,  P.V. Ramana Murty,  "The ideals of a bounded commutative BCK-algebra"  ''Math. Japon.'' , '''32'''  (1987)  pp. 723–733</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.S. Hoo,  "Injectives in the categories of BCK and BCI-algebras"  ''Math. Japon.'' , '''33'''  (1988)  pp. 237–246</TD></TR><TR><TD valign="top">[a3]</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">[a4]</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">[a5]</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">[a6]</TD> <TD valign="top">  K. Iséki,  S. Tanaka,  "Ideal theory of BCK-algebras"  ''Math. Japon.'' , '''21'''  (1976)  pp. 351–366</TD></TR><TR><TD valign="top">[a7]</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>
 +
 +
{{TEX|done}}

Latest revision as of 02:43, 15 February 2024

Algebras originally defined by K. Iséki and S. Tanaka in [a7] to generalize the set difference in set theory, and by Y. Imai and Iséki in [a5] as the algebras of certain propositional calculi. A BCK-algebra may be defined as a non-empty set $X$ with a binary relation $\ast$ and a constant $0$ satisfying the following axioms:

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

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

3) $x \ast x = 0$;

4) $x \ast y = 0$ and $y \ast x = 0$ imply $x = y$;

5) $x \ast 0 = 0$ implies $x = 0$;

6) $0 \ast x = 0$ for all $x$. A partial order $\le$ can then be defined by putting $x \le y$ if and only if $x \ast y = 0$. A very useful property is $(x \ast y) \ast z = (x \ast z) \ast y$.

A BCK-algebra is commutative if it satisfies the identity $x \ast (x \ast y) = y \ast (y\ast x)$ (cf. also Commutative ring). In this case, $x \ast (x \ast y) = x \wedge y$, the greatest lower bound of $x$ and $y$ under the partial order $\le$. The BCK-algebra is bounded if it has a largest element. Denoting this element by $1$, one has $1 \ast \{(1\ast x) \wedge (1\ast y)\} = x \vee y$, the least upper bound of $x$ and $y$. In this case, $X$ is a distributive lattice with bounds $0$ and $1$. A BCK-algebra is positive implicative if it satisfies the identity $(x\ast y)\ast z = (x\ast z) \ast (y\ast z)$. This is equivalent to the identity $x \ast y = (x\ast y)\ast y$. $X$ is called implicative if it satisfies the identity $x \ast (y \ast x) = x$. Every implicative BCK-algebra is commutative and positive implicative, and a bounded implicative BCK-algebra is a Boolean algebra.

An ideal of a BCK-algebra is a non-empty set $I$ such that $0 \in I$ and if $x \ast y \in I$ and $y \in I$ imply $x \in I$. The ideal is implicative if $(x\ast y) \ast z \in I$ and $y \ast z \in I$ imply $x \ast z \in I$. It is known that always $(x \ast z) \ast z \in I$. Note that in a positive implicative BCK-algebra, every ideal is implicative. Implicative ideals are important because in a bounded commutative BCK-algebra they are precisely the ideals for which the quotient BCK-algebras are Boolean algebras. Here, if $I$ is an ideal in a BCK-algebra, one can define a congruence relation in $X$ by $x \sim y$ if and only if $x \ast y \in I$ and $y \ast x \in I$. The set $X/I$ of congruence classes then becomes a BCK-algebra under the operation $[x]\ast [y] = [x\ast y]$, with $[0]$ as the constant and $[1]$ as the largest element if there exists a largest element $1$. Some, but not all, of the well-known results on distributive lattices and Boolean algebras hold in BCK-algebras, in particular in bounded commutative BCK-algebras. For example, the prime ideal theorem holds for bounded commutative BCK-algebras, that is, if $I$ is an ideal and $F$ is a lattice filter such that $I \cap F = \emptyset$, then there exists a prime ideal $J$ such that $I \subset J$ and $J \cap F = \emptyset$. Here, "prime ideal" simply means that if it contains $x \wedge y$, then it contains either $x$ or $y$.

Some of the homological algebra properties of BCK-algebras are known, see [a2]. There is also a close connection between BCK-algebras and commutative $l$-groups with order units (cf. $l$-group). Recall that an element $u$ in the positive cone $G_+$ of a commutative $l$-group $G$ is an order unit if for each $x \in G$ one has $x \le nu$ for some integer $n$. Let $G(u) = \{x \in G : 0 \le x < u\}$. For $x, y \in G(u)$, let $x \ast y = (x-y)_+ = (x-y) \vee 0$. Then $G(u)$ is a commutative BCK-algebra.

Fuzzy ideals of BCK-algebras are described in [a3] and [a4]. General references for BCK-algebras are [a6] and [a7].

References

[a1] C.S. Hoo, P.V. Ramana Murty, "The ideals of a bounded commutative BCK-algebra" Math. Japon. , 32 (1987) pp. 723–733
[a2] C.S. Hoo, "Injectives in the categories of BCK and BCI-algebras" Math. Japon. , 33 (1988) pp. 237–246
[a3] C.S. Hoo, "Fuzzy ideals of BCI and MV-algebras" Fuzzy Sets and Systems , 62 (1994) pp. 111–114
[a4] C.S. Hoo, "Fuzzy implicative and Boolean ideals of MV-algebras" Fuzzy Sets and Systems , 66 (1994) pp. 315–327
[a5] Y. Imai, K. Iséki, "On axiom systems of propositional calculi, XIV" Proc. Japan Acad. Ser. A, Math. Sci. , 42 (1966) pp. 19–22
[a6] K. Iséki, S. Tanaka, "Ideal theory of BCK-algebras" Math. Japon. , 21 (1976) pp. 351–366
[a7] K. Iséki, S. Tanaka, "An introduction to the theory of BCK-algebras" Math. Japon. , 23 (1978) pp. 1–26
How to Cite This Entry:
BCK-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BCK-algebra&oldid=16457
This article was adapted from an original article by C.S. Hoo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article