On

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Volume 2011, Article ID 676020,8pages doi:10.1155/2011/676020

Research Article

On

BE

-Semigroups

Sun Shin Ahn

1

_{and Young Hee Kim}

2

1_{Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea} 2_{Department of Mathematics, Chungbuk National University, Chongju 361-763, Republic of Korea}

Correspondence should be addressed to Young Hee Kim,[email protected]

Received 26 January 2011; Accepted 2 March 2011

Academic Editor: Young Bae Jun

Copyrightq2011 S. S. Ahn and Y. H. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The notion of aBE-semigroup is introduced, and related properties are investigated. The concept of leftresp., rightdeductive systems of aBE-semigroup is also introduced.

1. Introduction

Hu and Li, Is´eki and Tanaka, respectively, introduced two classes of abstract algebras:

BCK-algebras and BCI-algebras 1–3. It is known that the class of BCK-algebras is a proper

subclass of the class ofBCI-algebras. In1,4Hu and Li introduced a wide class of abstract

algebras:BCH-algebras. They have shown that the class ofBCI-algebras is a proper subclass

of the class of BCH-algebras. We refer to 5 for general information on BCK-algebras.

Neggers and Kim6introduced the notion of ad-algebra which is a generalization of

BCK-algebras, and also they introduced the notion of aB-algebra7,8, that is,Ix∗x 0,II

x∗0x,III x∗y∗zx∗z∗0∗y, for anyx, y, z∈X, which is equivalent to the idea

of groups. Moreover, Jun et al.9introduced a new notion, called anBH- algebra, which is

another generalization ofBCH/BCI/BCK-algebras, that is,I,II, andIVx∗y 0 and

y∗x 0 imply thatx y for anyx, y ∈ X. Walendziak obtained other equivalent set of

axioms for aB-algebra10. Kim et al.11introduced the notion of apre-Coxeter algebra

and showed that a Coxeter algebra is equivalent to an abelian group all of whose elements

have order 2, that is, a Boolean group. C. B. Kim and H. S. Kim12introduced the notion

of aBM-algebra which is a specialization ofB-algebras. They proved that the class of

BM-algebras is a proper subclass ofB-algebras and also showed that aBM-algebra is equivalent

to a 0-commutativeB-algebra. In13, H. S. Kim and Y. H. Kim introduced the notion of a

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an equivalent condition of the filter inBE-algebras. In14,15, Ahn and So introduced the

notion of ideals inBE-algebras and proved several characterizations of such ideals.

In this paper, by combiningBE-algebras and semigroups, we introduce the notion of

BE-semigroups. We define leftresp., rightdeductive systemsLDSresp., RDSfor short

of a BE-semigroup, and then we describe LDS generated by a nonempty subset in a

BE-semigroup as a simple form.

2. Preliminaries

We recall some definitions and results discussed in13.

Definition 2.1see13. An algebraX;∗,1of type2, 0is called aBE-algebra if

BE1x∗x1 for allx∈X,

BE2x∗11 for allx∈X,

BE31∗xxfor allx∈X,

BE4x∗y∗z y∗x∗zfor allx, y, z∈Xexchange.

We introduce a relation “≤” onXbyx≤yif and only ifx∗y1.

Proposition 2.2see13. IfX;∗,1is aBE-algebra, thenx∗y∗x 1 for anyx, y∈X.

Example 2.3see13. LetX:{1, a, b, c, d,0}be a set with the following table:

∗ 1 a b c d 0

1 1 a b c d 0

a 1 1 a c c d

b 1 1 1 c c c

c 1 a b 1 a b

d 1 1 a 1 1 a

0 1 1 1 1 1 1

2.1

ThenX;∗,1is aBE-algebra.

Definition 2.4see13. ABE-algebra X;∗,1is said to be self-distributive ifx∗y∗z

x∗y∗x∗zfor allx, y, z∈X.

Example 2.5see13. LetX:{1, a, b, c, d}be a set with the following table:

∗ 1 a b c d

1 1 a b c d

a 1 1 b c d

b 1 a 1 c c

c 1 1 b 1 b

d 1 1 1 1 1

2.2

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Note that theBE-algebra inExample 2.3is not self-distributive, sinced∗a∗0 d∗d

1, whiled∗a∗d∗0 1∗aa.

Proposition 2.6. LetXbe a self-distributiveBE-algebra. Ifx≤y, thenz∗x≤z∗yandy∗z≤x∗z

for anyx, y, z∈X.

Proof. The proof is straightforward.

3. BE

-Semigroups

Definition 3.1. An algebraic system X;,∗,1 is called a BE-semigroup if it satisfies the

following:

i X;is a semigroup,

ii X;∗,1is aBE-algebra,

iiithe operation “” is distributiveon both sidesover the operation “∗”.

Example 3.2. 1Define two operations “” and “∗” on a setX :{1, a, b, c}as follows:

1 a b c

1 1 1 1 1

a 1 1 1 1

b 1 1 1 1

c 1 a b c

∗ 1 a b c

1 1 a b c

a 1 1 b c

b 1 a 1 c

c 1 1 1 1

3.1

It is easy to see thatX;,∗,1is aBE-semigroup.

2Define two binary operations “” and “∗” on a setA:{1, a, b, c}as follows:

1 a b c

1 1 1 1 1

a 1 1 1 1

b 1 1 1 b

c 1 1 b c

∗ 1 a b c

1 1 a b c

a 1 1 b c

b 1 a 1 c

c 1 1 1 1

3.2

It is easy to show thatA;,∗,1is aBE-semigroup.

Proposition 3.3. LetX;,∗,1be aBE-semigroup. Then

i ∀x∈X 1xx11,

ii ∀x, y, z∈X x≤y⇒xz≤yz,zx≤zy.

Proof. iFor allx ∈ X, we have that 1x 1∗1x 1x∗1x 1 andx1

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iiLetx, y, z∈Xbe such thatx≤y. Then

xz∗yzx∗yz1z1,

zx∗zyzx∗yz11. 3.3

Hencexz≤yzandzx≤zy.

Definition 3.4. An elementa_/1in aBE-semigroupX;,∗,1is said to be a leftresp., right

unit divisor if

∃b/1∈X ab1 resp., ba1. 3.4

A unit divisor is an element ofXwhich is both a left and a right unit divisors.

Theorem 3.5. LetX;,∗,1be aBE-semigroup. If it satisfies the left (resp., right ) cancellation law

for the operation, that is,

∀x/1, y, z∈A xyyzresp., yxzx⇒yz, 3.5

thenXcontains no left (resp., right) unit divisors.

Proof. LetX;,∗,1satisfy the left cancellation law for the operationand assume thatx

y1 where_{x /}1. Thenxy1x1 byProposition 3.3i, which impliesy1. Similarly

it holds for the right case. Hence there is no leftresp., rightunit divisors inX.

Now we consider the converse ofTheorem 3.5.

Theorem 3.6. LetX;,∗,1be aBE-semigroup in which there are no left (resp., right ) unit divisors.

Then it satisfies the left (resp., right) cancellation law for the operation.

Proof. Letx, y, z∈Xbe such thatxyxzand_{x /}1. Then

xy∗zxy∗xz 1,

xz∗y xz∗xy1. 3.6

SinceXhas no left unit divisor, it follows thaty∗z1z∗yso thatyz. The argument is

the same for the right case.

Definition 3.7. LetX;,∗,1be aBE-semigroup. A nonempty subsetDofXis called a left

resp., right deductive systemLDSresp., RDS, for shortif it satisfies

ds1XD⊆Dresp.,DX⊆D,

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Example 3.8. LetX:{x, y, z,1}be a set with the following Cayley tables:

1 x y z

1 1 1 1 1

x 1 x 1 1

y 1 1 y z

z 1 1 z y

∗ 1 x y z

1 1 x y z

x 1 1 y z

y 1 1 1 z

z 1 1 1 1

3.7

It is easy to show thatX;,∗,1is aBE-semigroup. We know thatD:{1, x}is an LDS ofX,

butE:{1, y}is not an LDS ofX, sincezyz /∈Eand/ory∗x1∈E,y∈Ebutx /∈E.

LetX;∗,1be aBE-algebra, and leta, b∈X. Then the set

Aa, b:{x∈X|a∗b∗x 1} 3.8

is nonempty, since 1, a, b∈Aa, b.

Proposition 3.9. IfDis an LDS of aBE-semigroupX;,∗,1, then

∀a, b∈D Aa, b⊆D. 3.9

Proof. Letx∈Aa, bwherea, b∈D. Thena∗b∗x 1∈Dand sox∈Dbyds2. Therefore

Aa, b⊆D.

Theorem 3.10. Let{Di}be an arbitrary collection of LDSs of aBE-semigroupX;,∗,1, wherei

ranges over some index setI. Then∩i∈IDiis also an LDS ofA.

Proof. The proof is straightforward.

LetX;,∗,1be aBE-semigroup. For any subsetDofX, the intersection of all LDSs

resp., RDSs of X containing D is called the LDSs resp., RDSs generated by D, and is

denoted by D_l resp., D_r. It is clear that if D and E are subsets of a BE-semigroup

X;,∗,1satisfyingD ⊆E, thenDl ⊆ Elresp.,Dr ⊆ Er, and ifDis an LDSresp.,

RDSofX, thenDlDresp.,Dr D.

ABE-semigroupX;,∗,1is said to be self-distributive ifX;∗,1is a self-distributive

BE-algebra.

Theorem 3.11. LetX;,∗,1be a self-distributiveBE-semigroup and letDbe a nonempty subset of

Xsuch thatAD⊆D. ThenDl:{a∈X|yn∗· · ·∗y1∗a· · · 1 for somey1, . . . , yn∈D}.

Proof. Denote

B:a∈X|yn∗· · · ∗y1∗a

· · ·1 for somey1, . . . , yn∈D

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Leta∈Xandb∈B. Then there existy1, . . . , yn ∈Dsuch thatyn∗· · · ∗y1∗b· · · 1. It

follows that

1x1

xyn∗· · · ∗y1∗b

· · ·

xyn∗· · · ∗xy1

∗xb· · ·.

3.11

Sincexyi∈Dfori1, . . . , n, we have thatxb∈B. Letx, a∈Xbe such thata∗x∈Band

a∈B. Then there existy1, . . . , yn, z1, . . . , zm∈Dsuch that

yn∗· · · ∗y1∗a∗x

· · ·1, 3.12

zm∗· · · ∗z1∗a· · · 1. 3.13

UsingBE4, it follows from3.12thata∗yn∗· · · ∗y1∗x· · · 1, that is,a≤yn∗· · · ∗

y1∗x· · ·, and so from3.13andProposition 2.6it follows that

1zm∗· · · ∗z1∗a· · ·

≤zm∗· · · ∗z1∗

yn∗· · · ∗y1∗x

· · ·· · ·. 3.14

Thuszm∗· · · ∗z1∗yn∗· · · ∗y1∗x· · ·· · · 1, which impliesx∈ B. ThereforeBis

an LDS ofX. ObviouslyD ⊆B. LetGbe an LDS containingD. To showB ⊆G, letabe any

element ofB. Then there existy1, . . . , yn ∈Dsuch thatyn∗· · · ∗y1∗a· · · 1. It follows

fromds2thata∈Gso thatB⊆G. Consequently, we have thatDlB.

In the following example, we know that the union of any LDSsresp., RDSsDandE

may not be an LDSresp., RDSof a self-distributiveBE-semigroupX;·,∗,1.

Example 3.12. LetX:{1, a, b, c, d}be a set with the following Cayley tables:

1 a b c d

1 1 1 1 1 1

a 1 1 1 1 1

b 1 1 1 1 1

c 1 1 1 1 1

d 1 1 1 1 d

∗ 1 a b c d

1 1 a b c d

a 1 1 b b d

b 1 a 1 a d

c 1 1 1 1 d

d 1 1 b b 1

3.15

It is easy to check thatX;,∗,1is a self-distributive BE-semigroup. We know that D :

{1, a}andE:{1, b}are LDSs ofX, butD∪E{1, a, b}is not an LDS ofX, sinceb∗ca∈

D∪E,c /∈D∪E.

Theorem 3.13. LetDandEbe LDSs of a self-distributiveBE-semigroupX;·,∗,1. Then

D∪El:

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Proof. Denote

K:a∈X |x∗y∗a1 for somex∈D, y∈E. 3.17

Obviously,K ⊆ D∪El. Letb ∈ D∪El. Then there exist y1, . . . , yn ∈ D∪Esuch that

yn∗· · · ∗y1∗b· · · 1 byTheorem 3.11. Ifyi ∈Dresp.,Efor alli1, . . . , n, thenb∈D

resp.,E. Henceb∈Ksinceb∗1∗b 1resp., 1∗b∗b 1. If some ofy1, . . . , ynbelong

toDand others belong toE, then we may assume thaty1, . . . , yk∈Dandyk1, . . . , yn∈Efor

1≤k < n, without loss of generality. Letpyk∗· · · ∗y1∗b· · ·. Then

yn∗· · · ∗yk1∗p

· · ·

yn∗· · · ∗yk1∗

yk∗· · · ∗y1∗b

· · ·· · ·

1,

3.18

and sop∈E. Now letqp∗b yk∗· · · ∗y1∗b· · ·∗b. Then

yk∗· · · ∗y1∗q

· · ·

yk∗· · · ∗y1∗

yk∗· · · ∗y1∗b

· · ·∗b· · ·

yk∗· · · ∗y1∗b

· · ·∗yk∗· · · ∗y1∗b

· · ·

1,

3.19

which implies thatq∈D. Sincep∗q∗b q∗p∗b q∗q1, it follows thatb∈Kso that

D∪El⊆K. This completes the proof.

References

1 Q. P. Hu and X. Li, “On proper BCH-algebras,” Mathematica Japonica, vol. 30, no. 4, pp. 659–661, 1985.

2 K. Is´eki and S. Tanaka, “An introduction to the theory of BCK-algebras,” Mathematica Japonica, vol. 23, no. 1, pp. 1–26, 1978/79.

3 K. Is´eki, “On BCI-algebras,” Mathematics Seminar Notes, vol. 8, no. 1, pp. 125–130, 1980.

4 Q. P. Hu and X. Li, “On BCH-algebras,” Mathematics Seminar Notes, vol. 11, no. 2, pp. 313–320, 1983.

5 J. Meng and Y. B. Jun, BCK-Algebras, Kyung Moon Sa, Seoul, Korea, 1994.

6 J. Neggers and H. S. Kim, “On d-algebras,” Mathematica Slovaca, vol. 49, no. 1, pp. 19–26, 1999.

7 J. Neggers and H. S. Kim, “On B-algebras,” Matematichki Vesnik, vol. 54, no. 1-2, pp. 21–29, 2002.

8 J. Neggers and H. S. Kim, “A fundamental theorem of B-homomorphism for B-algebras,” International

Mathematical Journal, vol. 2, no. 3, pp. 207–214, 2002.

9 Y. B. Jun, E. H. Roh, and H. S. Kim, “On BH-algebras,” Scientiae Mathematicae, vol. 1, no. 3, pp. 347–354, 1998.

10 A. Walendziak, “Some axiomatizations of B-algebras,” Mathematica Slovaca, vol. 56, no. 3, pp. 301–306, 2006.

11 H. S. Kim, Y. H. Kim, and J. Neggers, “Coxeter algebras and pre-Coxeter algebras in Smarandache setting,” Honam Mathematical Journal, vol. 26, no. 4, pp. 471–481, 2004.

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13 H. S. Kim and Y. H. Kim, “On BE-algebras,” Scientiae Mathematicae Japonicae, vol. 66, no. 1, pp. 113–116, 2007.

14 S. S. Ahn and K. S. So, “On ideals and upper sets in BE-algebras,” Scientiae Mathematicae Japonicae, vol. 68, no. 2, pp. 279–285, 2008.

15 S. S. Ahn and K. S. So, “On generalized upper sets in BE-algebras,” Bulletin of the Korean Mathematical

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Discrete Mathematics

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Research Article

On

BE

-Semigroups

Sun Shin Ahn

and Young Hee Kim

1. Introduction

2. Preliminaries

3.

BE

-Semigroups

References

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Complex Analysis

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_{and Young Hee Kim}