数理解析研究所講究録
第 1846 巻 2013 年 35-4535

To BCI from Subtractive Algebra
井関清志 (Kiyoshi ISEKI)

After introducing the cocepts BCK, BCI (1966), manymathematicians have tried the generalizations of bothsystems. The final one(1994) is due to A. Ursini (Univ.
of Siena). It is a subtractive algebra[32]. This isan algebra with a constant and a binary operation*satisfying the following two axioms:$0$$x*x=0. x*0=x.$
In our discussion, there is another important identity,
which is very often used. Its identity is$0*x=0.$On the other hand, in BCI, the following facts hold.

1. $x*0=x,$$x*x=0,$2. $x\leq y\Rightarrow z*y\leq z*x,$ $x*z\leq y*z,$
3. $(x*y)*z=(x*z)*y$ , (permutation rule)
4. $(0*x)*(0*y)=0*(x*y)$ ,5. The minimal element of a branch with is given
by $0*(0*x)$ .
6. If is a minimal element, then $0*(0*x)=x.$
7. $x*y=y*x=0\Rightarrow x=y$ . (quasi-identity)
8. $0\leq x$ for all in BCK, but not in BCI.
$x$

$x$

$x$

36We have the atlas of proper BCI of order $n\leq$5(J.Hao[8], B. Hu[10]). J.Meng, Y.B. Jun and E.H.Roh[21]gave all proper BCI of order 6.
From now, we omi , so $x*y$ is denoted by $xy$$t^{*}$

The first generalization BCH of both systems isdue to Q.Hu. BCH is defined by the following conditions.

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Y.Komori(1984) introduced the concept of BCCalgebra:

1. $0x=0,$
2. $x0=x,$
3. $((xy)(zy))(xz)=0.$
4. $xy=yx=0\Rightarrow x=y.$This syestem was extensively studied by W.A. Dudek([3],[4],[5]and [6] .
Remark $((xy)(xz))(zy)=0$ and $xO=x\Rightarrow$$)$$(x(xy))y=0.$

From the first identity, $((xO)(xz))(zO)=0$ , Hence$xO=x\Rightarrow(x(xz))z=0$ . The identity obtained is
one of the axioms of BCK, BCI.
Y.B.Jun, $E,H$ ,Roh and H.S.Kim (1998) introduced
the concept of $BH$-algebra([17],[32]):

1$xx=0,$$x0=x,$

2.
3. $xy=yx=0\Rightarrow x=y.$
J.Neggers and H.S.Kim(2001) gave interesting examples of a $BH$ -algebra. Let
be the set of realnumbers.
$\mathbb{R}$$xy=\{\begin{array}{ll}0, if x=0,\frac{(x-y)^{2}}{x}, otherwise.\end{array}$

It is easy check thatis a $BH$ -algebra satisfying $Ox=0$ , but not a BCH-algebra. We know an
example of $BH$ in which Ox $=0$ does not hold.
$(\mathbb{R},$$0\rangle$$.,$

38. Neggers and H.S. Kim(1999) gave an algebra whichis called a -algebra[25].
$d$

1. $xx=0,$
2. $0x=0.$
3. $xy=0=yx\Rightarrow x=y.$The system includes a quasi-identity.
There is a simple example of a -algebra. Let X be
the set of non-negative real numbers. Define$d$$xy= \max\{0,x-y\}.$Then $X$ is a -algebra with $x0=x.$
Remark If a -algebra is associative, it is trivial.
$0=0x=(xx)x=x(xx)=x0$ Hence $x=0$ . So the-algebra is trivial. J.Negger and H.S.Kim [25].
$d$

$d$

$d$

They(2002) introduce a new concept of $B$ -algebra[30].

The axioms are
1. $xx=0,$
2. $x0=x,$
3. $(xy)z=x(z(0y))$ .
There is an interesting example of -algebra. Let$X$ be the set of all real numbers except for nonneg, Define $xy$ by
ative$B$- $n$$xy= \frac{n(x-y)}{n+y}$Then $X$ is a B–algebra. On the other hand, define$xy=xy^{-1}$ in a given group. Then is a Balgebra. M.Kondo and Y.B.Jun[21] proved that theclass of -algebra is same with the class of groups.
$G$

$B$

$G$

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However, J. Neggers and H.S. Kim work is never worthless(For example, ,Walendziak [34]).
C.B. Kim and H.S. Kim defined $BM$ algebra witha constant amd a binary operatio satisfying
1. $00=0,$
2. $x0=x,$
3. $(zx)(zy)=yx.$
$A$

$0$

$n^{*}$

Coxeter algabra is introduced by $H,S$ , Kim, Y.H.Kim
and J.Neggers(2004)
1. $xx=0,$
2. $x0=x,$
3. $(xy)z=(xz)y.$K.Iseki, J. Neggers and H.S. Kim defined -algebra.
$J$

1. $x*0=x,$
2. $x*(x*(y*(y*x)))=y*(y*(x*(x*y))$ .
Remark. In set theory, there is an identity:$A-(A-(B-(B-C)))=A\cap B\cap C.$

Axiom 2 is a special case of the above identity.
S. M. Hong, Y.B. Jun and M.Aliozturk(2003) definedan algebra which is called $gBCK$-algebra.

1. $x0=x,$
2. $xx=0,$
3. $(xy)z=(xz)y,$
4. $(xy)z=(xz)(yz))$

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A positive implicative BCK is $gBCK$ -akgebra, Thefollowing relations are true:
a $0x=0,$$)$b $(xy)x=0,$$)$

c $xy=0\Rightarrow(xz)(yz)=0.$$)$

Proof,$0=00=(xx)(xx)=(xx)x=0x.$$(xy)x=(xx)y=0y=0.$$xy=0\Rightarrow(xz)(yz)=(xy)z=0z=0.$

In 2005, Y.B. Jun and H.S.Kim introduced the concept of a substraction algebra. This is definedby the following axioms:
1. $x(yx)=x,$
2. $x(xy)=y(yx)$ ,3. $(xy)z=(xz)y.$
The algebra does not include a constant . However, $xx$ acts as a constantI introduced two systems in our early works. Oneof them is I-algebra which is formulated as follows:$0$$0.$

1. $x0=x,$
2. $0x=0,$
3. $((xy)(xz))(zy)=0,$
4. $(xy)(x(yx)\rangle=0,$
5. $xy=yx=0\Rightarrow x=y.$41

Until now, mentioned algebras and systems with
only one binary operation. On the hand, various
algebras and systens with several binary operations
are in the literature.
I introduced a system in our researches(1967). It
is Griss-algebra. The system has two operationsand $+.$
constant is included in Griss system.
$x\leq y$ means $xy=0$ . Griss system is formulated asfollows:
$I$

$*$

$A$

$0$

1. $0x=0,$
2. $x+x=x,$
3. $x;=y+x,$4, $(x+y)(x+z)\leq zy,$
5. $x\leq x+y,$
6. $xy=yx=0\Rightarrow x=y.$
J.MNeggers and H.S.Kim(2002) defined an algerbra with a constant and two binary operations satisfying the following three axioms.
$0$

1. $x0=x,$
2. $(0x)+x=0,$
3. $(xy)z=x(y+z)$ .
The algebra is called -algebra[32].
$\beta$There are other important generalizations of BCKand BCI due to Bucarest research group(under G. Georgescu).
They defined the concepts of pseudo- $MV$ , pseudo$BL$ , pseudo-BCK, Iseki algebra, and pseudo-Iseki algebras and a pseudo -norm. A.Iorgulescu(2003) defined many systems([12],[13]), for example, left-Iseki( $RP$ )
$t$

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algebra, , left H\’ajek(R)-algebra Among them,mentione only definiton of pseudo-BCK is given.
pseudo-BCK is a sytem, whereis a binary relation,
are binary operations, andis a constant,and is given by the following axiom
sysytem:$I$$\ldots$$A$$\mathbb{X}=(X, \leq,*,\Diamond, 0)$$\leq$$*,$$\Diamond$$0$

1. $(x*y)\Diamond(x*z)\leq z*y,$
2. $x*(x\Diamond y)\leq y,$ $x\Diamond(x*y)\leq y,$
3. $x\leq x,$
4. $0\leq x.$
5. $x\leq yy\leq x\Rightarrow x=y,$6.

$(x\Diamond y)*(x\Diamond z)\leq z\Diamond y,$$x\leq y\Leftrightarrow x*y=0\Leftrightarrow x\Diamond y=0.$

References

survey of BCK and BCI
algebras, Congresso de logica Aplicada a Tecnologia,LAPTEC 2000. 431-443.
[2] J.Allen, J.Eggers and H.S.Kim, -up and mirror algebras, Math. Japonica, $58(2004),381-388.$[3] W.A.Dudek, The number of subalgebras offinite BCC-algebras, Bll. inst. Math. Academia
Sinica. 20(1992), 129-136.
[4] W.A. Dudek, X. H.Zhang, On ideaJs and congruences in BCC-algebras, Czech. Math. $48(123)(1998)$ ,21-29.
[5] $W,A$ ,Dudek and X.H.Zhang, On atoms in BCCalgebras, Diss. Math. 15(1995), 81-85.
[1] M.Abe and K.Iseki,

$A$

$L$

$J.$

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[29] J.Neggers, Y.B.Jun and H.S.Kim, On -idealsin -algebras, Math. Slovaca, 49(1999), 243-251.
[30] J.Neggers and H.Kim, On -algebras, Mat.
Vesnik 54(2002), 21-29.
[31] J.Neggers and H.S. Kim, On -algebras, Math.
Slovaca, $52(2002),517-530.$[32] J.Neggers and $H,S$ ,KIm, On analytic -algebras,Math,Japonica, 53(2001), 157-163.
[33] A. Ursini, On substractive varieties, I-IV, Alg.
Universalis, 31(1994), 204-222, 36(1996), 222-259, 37$(1997),296-333.38(1997),355-389.6[34]$ A.Walendziak,
A Note on normal subalgebras in -algebras, Sci.
Math. Japonicae, 62(2005), 49-53.
$d$

$d$

$B$

$\beta$$T$$B$