数理解析研究所講究録 1222 巻 2001 年 40-45

40Topics on finite and countable infinite BCKKiyoshi ISEKI(井関 清志)

In this Note, we present some new results on BCK without proofs, inparticular, we mainly concern with finite and countable infinite BCK.
First Igive anew de finition of BCK wlu\dot & is equivalent to an old definbe apartialy ordered set with ation(for xmple, see [5], [7], [8]). Let
least element 0on which abinary operaion *is defined. We assume that theis reflexive, antisy mmetric and transitive.
partial order
We say that is aBCK if binary operation * on satisfies the following
conditions:$P$$\mathrm{e}$$x$$\leq y$$P$$P$1) $(x *y)*(x *z)$ $\leq z$ $*y$ ,,2)) $*(x *y)$3) $x*x=0$ ,4) $0*x$ $=0$ ,5) $*y=0$ is equivalent to $x\leq y$ .
$\leq y$$x$$x$

We know some special classes of BCK which are defined by the following
way: Let $X$ be aBCK.
(1) $X$ is positive implicative, if $(x *y)*y=x$ $*y$ for any , in $X$ .
(2) $X$ is commutative, if $*(x *y)=y*(y*x)$ for any , in $X$ .
$(x *y)$ is the greatest lower bound of $x,y$ . Hence it is denoted
Then. Therefore acommutative BCK is alower semilattice.
by(3) $X$ is implicative, if $X$ is positive nplicat\’ive and commutative.
(4) $X$ is aBCK with condition (S), if for any $a,b\in X$ , non-em pty set
$x$

$x$

$x$

$y$

$y$

$x$ $*$$x$$\Lambda y$$\dot{\mathrm{u}}$$\{x :X *a\leq b\}$.

has the gereatest element in $X$ . Its element is denoted by $aob$
Then is partially ordered commutative semigroup with respect to thejust introduced operation . This is refered as the associated (p.o.commutative)
semigroup of $X$ .
The operation has the following basic properties:
6) $x,y$ $\leq xoy$ $=yox$ ,7) so(yoz) $=(x\psi)oz$ ,8) $*(y*z)$ $=x$ $*(yoz)$ ,9) $\leq yarrow xoz$ $\leq yoz$ for any $\in X$ .
We can find many basic and useful properties of these classes in [7] and[8]. Among them, we only mention some results which hold in BCK$\mathrm{X}$

$0$

$0$

$x$
$x$

$z$

4110) $x*0=x$ ,11) $(x*y)*z=(x*z)*y$ . (permutation rule)12) $x*y\leq x_{:}$
13) $x*[x*(x*y)]=x*y$ .
(5) W.H.Cornish introduced aconcept which is called aBCK with supremunl [4],.
Let be aBCK with conidition (S). is called aBCK with supremum, if the following identity holds for any elements , , in $X$ :$($$[_{9}^{r}])$$\mathrm{X}$$\mathrm{X}$

$x$

$y$

$z$

$xo(y*x)=yo(x*y)$ .
$xo(y*x)$ is denoted by $x\vee y$. It is the least upper bound of $x,y$ . Therefore,aBCK with supremum is an upper semilattice.
Remark 1. If the conidtion 4) in the definition of BCK is replaced by acondition: there is no elements less than 0, namely $0\leq x$ implies $x=0$ , weobtain the concept of BCI.
Theorem 1. Thereat least one $BCK$ structure on any partially ordered set with a least element 0. This $BCK$ structure is given by the following
way:0, if $x\leq y$ ,$x*y=\{$, otherwise.
$e\dot{m}b$$x$

Consequently this structure is positive implicative, as easily seen. Then
we can not change $x(=x*y)$ into greater than . In this sense, the obtained
BCK is maximal, namely if there is anew BCK structure on $X$ , then $x*y$
can not greater than $x(=x*y)$ in the first given BCK structure.
To define another important class of BCK, we introduce asimple concept
on apartially ordered set.
Definition 1. Apartially ordered set with aleast element 0is calledthe typeif there exists an element $a(\neq 0)$ such that $a\leq x$ for every
non-zero element in $P$.
Let us denote by 1such element .
Avery simple BCK is afan shaped one which has two different kinds.
We define two kinds of fans, namely Japanese style and Chinese style. Oneof them is of the type Y.
Definition 2. Apartially ordered set $P$ with aleast element 0is of aJapanese (style) fan if it is of type a1ld for any $a(\neq 0,1)$ in $P$, there areno elements , satisfying $1<x<a,a<y$ . $P$ is called Chinese (style)
fan if it is not atype and for any $a(\neq 0)$ there are no elements $x,y$ suchthat $0<x<a,a<y$ .
$z$

$x$

$P$

$\mathrm{Y}$

$x$

$a$

$\mathrm{Y}$

$x$

$y$

$\mathrm{V}$42
In the theory of BCK, the distinctions are quite important. It is very easy
to introduce at least one BCK (maximal) structure by applying Theorem 1.
lese style fan and
There exists no other BCK(maximal) structure on anyaU subsets including 0are subalgebras. On the other hand, any Japanesestyle fan has some other BCK structures.
Problem 1. Find all(finite and infinite) BCK with only one BCK structurae. Are there none in the finite BCK with only one structure except theChinese style one ?The answer may depend on the order.
Example 1. Let us consider the Japanese style fan of order 5as anexample. Then the -table of the maximal BCK is given in the following leftcase, there exist more three different BCK strcutures on it.
tables. In$\mathrm{C}1\dot{\mathrm{m}}$$*$$\mathrm{t}\underline{\mathrm{h}\mathrm{i}}\mathrm{s}$1 2 3 40000000 0 0 0 0 0$*$0 1 2 3 4$*$$0$110000110000

2220223333034 4 4 4 4 0

2 2 1 0 1 1
3 3 1 1 0 1
4 4 1 1 1 0

The above right structure is commutative. Other two structures are mentioned in the following tables, but it seems that no these structures havespecial kinds of properties.
$*$

$0$

$1$

$2$

$3$

$4$

$*$

$0$

$1$

$2$

$3$

$4$

000000000000
110000110000
221011221010
3 3 1 1 0 1333303
4 4 4 4 4 0

4 4 4 4 4 0

All elements in aBCK $X$ appear in the column containing 0 in the *table of $X$ . All elements which appear in the diagonal of tlne -table are 0.
Let us consider the part of the triangle of the left side of the diagonal, and(the 0-column). For example, &0m the above firstelminate the ffist coltwo *tables, we have the following parts:$*$$\mathrm{l}\mathrm{u}\mathrm{m}$213311
4 4 4

1 1 1

Such atriangle is called the essential part ofX. Similarly, we can define
the quasiessential part of X. Let us consider the triangle formed through the43
right side of the diagonal. We elminate the first rowow) of the triangle.
The triangle obtained is called the quasiesential part of X.
The quasiessential parts of the first two in Example 1are as follows:$(0\ovalbox{\tt\small REJECT}$

0000002211
31
For any BCK linearly ordered, all elements of its quasiessential part are0, as easily seen.
Definition 3. ABCK is called minimal, if aU elements of the essential
and quasiessential parts are 0or 1.
The above Japanese style fan has aminmal structure, but for aChinese
style fan, it is not always true. Then we have the following importantTheorem 2. A BCIf is the type if and only if it is minimal.
Hence any linear(finite or infinite) BCK has aminimal structure. Theminimal lnear BCK of order 3is ommutative, but such aBCK of order 4
is not commutative.
The essential parts of the later two structures are$\mathrm{Y}$$\mathrm{c}$1133114 4 44 4 4The first element of these essential parts are 1. They are smaller than
the first element 3of the maximal structure, and greater than the first element 1of the minim al structure. There are the same situations among
the corresponding elements. The same fact holds for the quasiessential part.
Therefore we can say that each structure is between the maximal and minimal structures(This is valid for anoncase). For every finite BCK,
we can theoretically find all BCK structures, but the calculation is not easy.
This is very tedious work.
Problem 2. Find an algorithm to determine all finite BCK structures.
Is there aTuring machine to describe the structures of all finite BCK ?
Next we consider afactor(branch) of apartiaUy ordered set $P$ with aleast
element 0.
Definition 4. Each member of familyof subsets of aBCK $X$ iscalled factor(branch) if it satisfies the following conditions:(F-1) Eachis apartially ordered set with 0,(F-2) $X$ is the union of,(F-3) $P_{m}\cap P_{n}=\{0\}$ for $m\neq n$ .
$\mathrm{m}\mathrm{i}\mathrm{r}\dot{\mathrm{u}}\mathrm{m}\mathrm{a}\mathrm{l}$$\{P_{n}\}$$P_{n}$$\{P_{n}\}$44
Then each is aBCK, and for $x\in P_{m}$ , $y\in P_{n}(m\neq n)$ we have $x*y=x$ .
be afamily of partially ordered sets. Let us supConversely, letis adisjoint
ment 0 is common in aU , andpose that the least
has aBCK
family except the element 0. Moreover, we suppose each ofstructure. Then the following result holds true.
Theorem 3. On the union $P$ of Pn, a BCIf structure is uniquely intrOis subalgebra under the structure on $P$ .
duced, and each
For $x\in P_{m}$ and $y\in P_{n}(m\neq n)$ ,$P_{n}$$\{P_{n}\}$$P_{n}$$\mathrm{e}\mathrm{l}\mathrm{e}$$\{P_{n}\}$$\{P_{n}\}$$P_{n}$$x*y=x$ .

is preserved in $P$ .
The BCK struc rure of each
For example, the union of several two elements BCK is aCliese style
fan and the BCK structure is given by the above equation. The uniquenessof the BCK structure on aChinese style fan also follow from Theorem 3.
Moreover, $X$ is positive implicative(commutative) if and if so is eachSo, each
is implicative if and if $P$ is implicative. Many interesting
commutative BCK are constructed by the above method. But acommutative
BCK is very strange in BCK theory. Its structure has not completely known
until now.
, there exist amaximal BCK, a minimal BCK, aFinally, as easily
positive implicative BCK, acommutative BCK, an imlcative BCK with any
fact is very useful to study various
cardinality. Iwold like to mention
classes of BCK. In this Note, we considered the case of the union of BCK
such that only 0is identified. An important researche program is to develop
general identification theory of BCK.
$P_{n}$$P_{m}$$P_{m}$$\mathrm{s}\mathrm{e}\mathrm{n}$$\mathrm{t}\mathrm{l}\dot{\mathrm{u}}\mathrm{s}$References[1] M.Abe and K.Iseki, Asurvey on BCK and BCI-algebras, Anais doCongresso de Logica Aplicada aTecnologia, LAPTEC 2000, 431-443.
[2] J.M.Harper and J.E.Rubin, Variations of Zorn’s lemma, principles ofcofinality, and Hausdorff’s maximal principle I, II, .Notre Dame Journal of, 161-163.
Formal logic, 17(1976),[3] W.H.Cornish, BCK-algebras with supremum, Math. Japonica, 27(1982),$\mathrm{i}$$56_{\mathrm{d}-\theta 88}^{rr},18(1977)$

63-73.

[4] W.H.Cornish, BCK-algebras with asupremum II. Distributivity andinterpolation, Math. Japonica, 29(1984), 439-337.
[5] K.Iseki, Away to BCK and related systems, Math.Japonica, 52(2000),
163-170.
[6] K.Iseki, On finite BCK with condition (S), Proc. Algebra, Languages
and Computation, 2000, 16-25

45[7] K.Iseki azxd Sh.Tanaka, An in troduction to thetheory of BCK-algebras,
Math. Japonica, 23(1978), 1-21.
[8] J.Meng and Y.B.Jun, BCK-algebras, Ky ung Moon Ss. Co. 1994.

Correction of my paper On finite BCK with condition (S) in Proc.
4th Sy mposium on Algebraic Langueges and C omputation(2000). 1. p.16.
15 $x*(y*x)$ should be read as $x*(y*z)$ .
2. p.18. In the right table in 7, x2i $=2$ should be read $x_{21}=3$ .
3. p.18. In tlle right table in 10, $x_{11}=2$ and $x_{12}=3$ should be respectively read as $x_{11}=1$ and $x_{12}=2$ .
4. p.18. In the right table in 6, $x_{11}=2$ should be read as $x_{11}=1$ .
5. p.18. in the right table in 13, X22 $=3$ should be read as $x_{22}=2$