Some Aspects of d-Units in d/BCK-Algebras

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Volume 2012, Article ID 141684,10pages doi:10.1155/2012/141684

Research Article

Some Aspects of d-Units in d/BCK-Algebras

Hee Sik Kim,

¹

J. Neggers,

²

and Keum Sook So

³

1Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea

2Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA

3Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea

Correspondence should be addressed to Keum Sook So,[email protected] Received 13 August 2012; Revised 2 November 2012; Accepted 9 November 2012 Academic Editor: Frank Werner

Copyrightq2012 Hee Sik Kim et al. 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.

We explore properties of the set of d-units of ad-algebra. A property of interest in the study of d-units in d-algebras is the weak associative property. It is noted that many other d-algebras, especially BCK-algebras, are in fact weakly associative. The existence ofd/BCK-algebras which are not weakly associative is demonstrated. Moreover, the notions of ad-integral domain and a left-injectivity are discussed.

1. Introduction

Is´eki and Tanaka introduced two classes of abstract algebras: BCK-algebras and BCI-algebras 1, 2. It is known that the class of BCK-algebras is a proper subclass of the class of BCI- algebras. Neggers and Kim introduced the notion of d-algebras which is another useful generalization of BCK-algebras and then investigated several relations betweend-algebras and BCK-algebras as well as several other relations betweend-algebras and oriented digraphs 3. After that some further aspects were studied4–7. As a generalization of BCK-algebras, d-algebras are obtained by deleting two identities. Thus, one may introduce an additional operationand replace one of the deleted new algebras, for example,x∗x∗y∗y 0, to obtain new algebras X,∗,,0 for which the conditions i xy∗x∗ y 0 and ii z∗x∗y 0 impliesz∗xy 0, yielding a companiond-algebra which shares many properties of BCK-algebras and such that not everyd-algebra is one. Allen et al.4 developed a theory of companiond-algebras in suﬃcient detail to demonstrate considerable parallelism with the theory of BCK-algebras as well as obtaining a collection of results of a novel type. Recently, Allen et al. 8 introduced the notion of deformation in d/BCK- algebras. Using such deformations they constructedd-algebras from BCK-algebras in such a manner as to maintain control over properties of the deformed BCK-algebras via the nature of

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the deformation employed and observed that certain BCK-algebras cannot be deformed at all, leading to the notion of a rigidd-algebra and consequently of a rigid BCK-algebra as well.

In this paper we study properties ofd-units ind-algebras X,∗,0, that is, elements xofXsuch that x∗X X. Since 0∗X {0},|X| ≥ 2 implies 0 is not ad-unit ofX,∗,0.

Hence, d-algebras X,∗,0 such that every non-zero element is ad-unit are special in the sense that they are “complete” with respect to this property. They are also not uncommon seeProposition 3.2. It turns out that the property of weakly associativity ind-algebras is an important property in this context. In addition, we consider the class ofd-integral domains and left-injective elements ofd-algebrasdefined belowin analogy with the usual notions in the theory of rings and their modules, where again thed-units investigated in this paper also play a significant role.

2. Preliminaries

In this section, we introduce some notions and propositions ond-algebras discussed in3,8–

10for reader’s convenience.

Anordinaryd-algebra3is an algebraX,∗,0where∗is a binary operation and 0∈Xsuch that the following axioms are satisfied:

Ix∗x0, II0∗x0,

IIIx∗y0 andy∗x0 implyxyfor allx, y∈X.

For brevity we also callX ad-algebra. InX we can define a binary relation “≤” by x≤yif and only ifx∗y0.

A BCK-algebra1is ad-algebraXsatisfying the following additional axioms:

IV x∗y∗x∗z∗z∗y 0,

V x∗x∗y∗y0 for allx, y, z∈X.

Example 2.1see3. aEvery BCK-algebra is an ordinaryd-algebra.bLetR be the set of all real numbers and definex∗y:x·x−y,x, y∈R, where “·” and “−” are the ordinary product and subtraction of real numbers. Thenx∗x0,0∗x0, x∗0x². Ifx∗yy∗x0, thenx·x−y 0 andy·y−x 0, that is,x0 orxyandy0 oryx, that is, x0 andy 0orx 0 andyxorx yandy0orxyandyx; all imply xy. Hence,R;∗,0is an ordinaryd-algebra. But it is not a BCK-algebra, since axiomV fails:2∗2∗0∗0 16/0.

An algebra X;∗,0, where∗is a binary operation and 0 ∈ X, is said to be a strong d-algebra9if it satisfiesI,IIandIII^∗for allx, y∈X, where

III^∗x∗yy∗ximpliesxy.

Obviously, every strongd-algebra is ad-algebra, but the converse needs not be true. Thed- algebra inExample 2.1bis not a strongd-algebra, sincex∗yy∗ximplies eitherxyor x−y.

Example 2.2see9. LetR be the set of all real numbers and definex∗y: x−y·x−ee, x, y, e∈R, where “·” and “−” are the ordinary product and subtraction of real numbers. Then x∗xe;e∗xe;x∗yy∗xeyieldsx−y·x−e 0,y−x·y−e 0 andxy orxey, that is,xy, that is,R;∗, eis ad-algebra.

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However,R;∗, eis not a strongd-algebra. Ifx∗yy∗x⇔x−y·x−ee y−x·

y−ee⇔x−y·x−e −x−y·y−e⇔x−y·x−ey−e 0⇔x−y·xy−2e 0⇔ x yorxy 2e, then there existx eαandy e−αsuch thatxy 2e, that is, x∗yy∗xandx /y. Hence, axiomIII^∗fails and thus thed-algebraR;∗, eis not a strong d-algebra.

Theorem 2.3see11. The following properties hold in a BCK-algebra: for allx, y, z∈X, B1x∗0x,

B2 x∗y∗z x∗z∗y.

A BCK-algebraX,∗,0is said to be bounded if there exists an elementx0∈Xsuch that x≤x0for allx∈X. We denote it byX,∗,0, x0. Note that the usual notation is 1 rather thanx0

in literatures. We call such an element the greatest element ofX. In a bounded BCK-algebra, we denotex0∗xbyNx. A bounded BCK-algebraX,∗,0, x0is called a BCK_DN-algebra12 if it verifies conditionDN:

DNNNxxfor allx∈X.

A BCK-algebraX,∗,0is said to be commutative ifx∗x∗y y∗y∗xfor any x, y∈X. We refer useful textbooks for BCK/BCI-algebra to11–14.

Theorem 2.4. IfX,∗,0is a bounded commutative BCK-algebra, thenNNxxfor anyx∈X.

It is well known that bounded commutative BCK-algebras, D-posets and MV- algebras are logically equivalent each othersee13, Page 420.

Definition 2.5see10. LetX,∗,0be ad-algebra and∅/I ⊆X.I is called ad-subalgebra ofXifx∗y∈Iwheneverx∈Iandy∈I.Iis called a BCK-ideal ofXif it satisfies:

D00∈I,

D1x∗y∈Iandy∈Iimplyx∈I.

Iis called ad-ideal ofXif it satisfiesD1and

D2x∈Iandy∈Ximplyx∗y∈I, that is,I∗X⊆I.

Note that, by axiomIand definition ofd-subalgebra, 0∈Xcan be deduced easily.

Example 2.6see10. LetX :{0,1,2,3,4}be ad-algebra which is not a BCK-algebra with the following table:

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

2.1

ThenI :{0, a}is ad-ideal ofX.

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In ad-algebra, a BCK-ideal need not be ad-subalgebra, and also ad-subalgebra need not be a BCK-ideal. Clearly,{0}is ad-subalgebra of anyd-algebraXand everyd-ideal ofX is ad-subalgebra10.

LetXbe ad-algebra andx∈X. Definex∗X :{x∗a|a∈X}.Xis said to be edge if for anyx∈X,x∗X{x,0}. It is known that ifXis an edged-algebra, thenx∗0xfor any x∈X3.

3. d -Units and Weakly Associativity

Let X,∗,0 be ad-algebra. An element xof X is said to be a d-unit ifx∗X X, where x∗X:{x∗y|y∈X}.

Example 3.1. LetX:{0,1,2,3}be a set with the following table:

∗ 0 1 2 3 0 0 0 0 0 1 1 0 3 2 2 2 1 0 3 3 3 3 3 0

3.1

ThenX,∗,0is ad-algebra. It is easy to show that 1, 2 ared-units ofX. Proposition 3.2. LetX,,·,0,1be a field. Define a binary operation “∗” onXby

x∗y:x· x−y

3.2

for anyx, y∈X. ThenX,∗,0is ad-algebra such that every non-zero element ofXis ad-unit.

Proof. It is easy to show that X,∗,0is ad-algebra. Given a non-zero element x and any elementuinX, the equationx∗yx²−xyuhas a solutiony x²−u/x.

Note that thed-algebraX,∗,0inProposition 3.2is not a BCK-algebra, sincex∗0 x²/x.

Ad-algebraX,∗,0is said to be weakly associative if for anyx, y, z∈X, there exists aw∈Xsuch thatx∗y∗zx∗w.

It is known that ifX,∗,0is a BCK-algebra with conditionS, thenx ∗ y ∗ z x∗ y◦ zfor allx, y, z∈X, wherey◦zis the greatest element of the setAy, z:{x∈X| x∗y≤z}11. Hence every BCK-algebra with conditionSis weakly associative.

Proposition 3.3. Thed-algebraX,∗,0defined inProposition 3.2is weakly associative.

Proof. Givenx, y, z∈X, we letw:x x−yz−xx−y². Then we have x∗wxx−w x

x−y x

x−y

−z

x∗y

x∗y−z

x∗y

∗z, 3.3

proving the proposition.

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∗ 0 1 2 3 0 0 0 0 0 1 1 0 3 2 2 3 3 0 1 3 2 1 2 0

3.4

ThenX,∗,0is ad-algebra which is not a BCK-algebra. We know that3∗1∗23, but there is no elementw∈Xsuch that 3∗w3. HenceX,∗,0is not weakly associative.

Proposition 3.5. LetX,∗,0be a weakly associatived-algebra andx, y ∈ X. Ifx∗yis ad-unit, thenxis also ad-unit.

Proof. Letu∈Xbe an arbitrary element ofX. Thenx∗y∗zufor somez∈X. SinceXis weakly associative, there existsw∈Xsuch thatx∗y∗zx∗w, provingux∗w, which shows thatxis ad-unit.

Proposition 3.6. LetX,≤be a poset with minimal element 0. Define a binary operation “∗” onX by

x∗y

0 ifx≤ y,

x otherwise. 3.5

ThenX,∗,0is a weakly associative BCK-algebra.

Proof. Givenx, y, z ∈ X, we have either x∗y∗z 0 or x∗y∗z x. Now, assume x∗y∗z 0. SinceX is a BCK-algebra,x∗y ≤ xand hencex∗y∗x 0. If we take w:x, then we obtainx∗y∗z0x∗x. Assumex∗y∗zx. If we takew:0, then x∗wx∗0x X∗y∗z, provingXis weakly associative.

Note that the BCK-algebra X,∗,0 discussed in Proposition 3.6 is a dual Hilbert algebra see 12, Page 30. By routine calculations, we found that there is no weakly associative BCK-algebras with order≤6.

By Propositions3.3and3.6, the class of weakly associative BCK-algebras is a proper subclass of weakly associatived-algebras.

Let X,∗,0 be a d-algebra and let N : {0,1,2,3, . . .}. We define the set of all monomialsaz^k,a /0, k ∈ N and 0 : {0z^k}_k∈N byMX wherez is an indeterminate. We may regard 0≡0z^kfor allk∈N.

Define a binary operation “” onMXby

az^k bz^l: a∗bz^αkβl, 3.6

whereαandβare fixed elements ofN. Then we obtain the following.

Proposition 3.7. IfX,∗,0is ad-algebra and ifα /βinN, thenMX, ,0is ad-algebra.

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Proof. For anyaz^k, bz^l ∈MX, we haveaz^k az^k a∗az^αβk 0z^αβk0 and 0 bz^l 0z^k bz^l 0∗bz^αkβl0z^αkβl0.

Assume thataz^k bz^lbz^l az^k0. Thena∗bb∗a0 andαkβlαlβk. Since X,∗,0is ad-algebra, we obtainabandα−βk α−βl. It follows that eitherαβor kl. Sinceα /β, we haveabandkl, that is,az^kbz^l. This proves the proposition.

UsingProposition 3.7, we construct a BCK-algebra which is not weakly associative.

Example 3.8. Define a binary operation “∗” on X : 0,1byx∗y : max{0, x−y}for all x, y∈X. Then it is easy to see thatX,∗,0is a BCK-algebra. If we takeα:3,β:7, then by Proposition 3.7,MX, ,0is ad-algebra, whereaz^k bz^l: a∗bz^3k7lfor allaz^k, bz^l∈MX.

We claim thatMX, ,0is not weakly associative. By routine calculation, we obtain 1/2z³1/3z^l1/12z^m 1/12z^2773lmand1/2z³ Azⁿ 1/2−Az⁹⁷ⁿfor any Azⁿ ∈ MX. IfMX, ,0is weakly associative, then1/12z^2773lm 1/2−Az⁹⁷ⁿ for someAzⁿ ∈MX. It follows thatA5/12 and 187n−m−3l, and hence 7 divides 18, a contradiction.

We claim thatMX, ,0is a BCK-algebra. For anyaz^k, bz^l, czⁿ∈MX, we haveaz^k az^k bz^l bz^l a∗a∗b∗bz^33k73k7l7l 0z^33k73k7l7l 0 andaz^k bz^l az^k czⁿczⁿ bz^l a∗b∗a∗c∗c∗bz^q 0z^q 0 for someq∈ N. Hence MX, ,0is a BCK-algebra which is not weakly associative.

4. d -Units in BCK-Algebras

Proposition 4.1. IfX,∗,0is a BCK-algebra andx0is ad-unit ofX, thenX,∗,0, x0is a bounded BCK-algebra.

Proof. Letx0be ad-unit ofX. Thenx0∗X X, which means that for anyy∈X, there exists u∈Xsuch thatyx0∗u. Hence we havey∗x0 x0∗u∗x0 x0∗x0∗u0∗u0 for anyy∈X. This proves thatX,∗,0, x0is a bounded BCK-algebra.

The converse ofProposition 4.1need not be true in general.

Example 4.2. Consider the BCK-algebraX,∗,0 11, Page 252with the following table:

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

4.1

ThenX,∗,0,4is a bounded BCK-algebra not verifying conditionDN, but 4 is not ad-unit, since 4∗X{0,2,4}/X.

Proposition 4.3. LetX,∗,0, x0 be a BCK_DN-algebra and let x0 ∈ X. Then x0 is the greatest element ofXif and only ifx0is ad-unit ofX.

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Proof. Letx0be the greatest element ofX. SinceXis a BCK_DN-algebra, byTheorem 2.4, we havexx0∗x0∗x∈x0∗Xfor anyx∈X, that is,X⊆x0∗X, proving thatx0is ad-unit of X. The converse was proved inProposition 4.1.

Example 4.4. Consider the BCK-algebraX,∗,0with the following table:

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

4.2

ThenX,∗,0,4is a BCK_DN-algebra11, Page 253, namely a bounded commutative BCK- algebra, and 4 is both the greatest element ofXand ad-unit ofX.

5. d -Integral Domains

LetX,∗,0be ad-algebra. An elementx∈Xis said to be ad-zero divisor ofXif there exists an elementy/xinXsuch thatx∗y0.

∗ 0 1 2 3 0 0 0 0 0 1 1 0 3 0 2 2 2 0 1 3 2 3 0 0

5.1

ThenX,∗,0is ad-algebra and 0,1,3 ared-zero divisors ofX.

Note that if|X| ≥2, then 0 is ad-zero divisor ofX.

LetX,∗,0be a BCK-algebra. Ifx∗y0 withy /x, thenx < yin the induced order, that is,x is not a maximal element ofX. This shows that every non-maximal element of a BCK-algebraXis ad-zero divisor ofX.

Ad-algebraX,∗,0is said to be ad-integral domain if every non-zero elementxis not ad-zero divisor, that is,x /0, x∗y0, y∈Ximpliesxy.

Proposition 5.2. Thed-algebraX,∗,0inProposition 3.2is ad-integral domain.

Proof. Ifx∗yxx−y 0 andx /0, thenx−y0, that is,xy. Hencexis not ad-zero divisor and the conclusion follows.

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∗ 0 1 2 3 0 0 0 0 0 1 1 0 1 1 2 2 2 0 1 3 2 3 3 0

5.2

ThenX,∗,0is ad-integral domain which is not a BCK-algebra, since every non-zero element is not ad-zero divisor.

Given two posetsXandY, we construct the ordinal sumX⊕Y ofXandY ifx≤yfor allx∈Xandy∈Y 15.

Proposition 5.4. Let X,≤,0be an ordinal sumX {0} ⊕A, where Ais an anti-chain. If one defines a binary operation “∗” onXby

x∗y

0 ifx≤ y,

x otherwise, 5.3

then the BCK-algebraX,∗,0is ad-integral domain.

Proof. Ifx /0 inX, then x ∈ A. SinceX is a BCK-algebra, we havex ∗ 0 x, and ify ∈ A, y /x, thenx∗yx. Hencex /0, x∗y0 impliesyx, proving the proposition.

In the situation ofProposition 5.4, it follows thatx /0 is ad-unit if |A| 1. Indeed, x∗X {0, x}for all x ∈ A, andx∗X X implies|X| 2. If |X| ≥ 3, then X,∗,0is a d-integral domain having nod-units.

6. Left-Injective

LetX,∗,0be ad-algebra. A non-zero elementx∈Xis said to be left-injective ifx∗yx∗z for ally, z∈Ximpliesyz. Ad-algebraX,∗,0is said to be left-injective if every non-zero element ofXis left-injective.

Proposition 6.1. LetX,∗,0be ad-algebra and|X| ≥ 2. Ifx ∈Xis left-injective, then it is not a d-zero divisor ofX.

Proof. If we assume thatxis ad-zero divisor ofX, thenx∗y0 for somey/xinX. SinceX is ad-algebra,x∗x0x∗y. It follows fromxis left-injective thatxy, a contradiction.

Proposition 6.2. LetX,∗,0be a finited-algebra andx∈X. Ifxis left-injective, then it is ad-unit.

Proof. Given a left-injective elementx, we define a mapϕx :X → Xbyϕxa:x∗a. Then it is injective mapping, sincexis left-injective. SinceXis finite,ϕxis onto, which proves that x∗XϕxX X. This proves thatxis ad-unit.

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In the infinite case,Proposition 6.2need not be true. We give an example ofd-algebra such that every non-zero elementx∈Xis a left-injective, but not ad-unit element.

Example 6.3. LetX : R be the set of real numbers and letx∗y : tan⁻¹xx−yfor any x, y∈X. Then it is easy to show thatX,∗,0is ad-algebra which is not a BCK-algebra, since x∗0 tan⁻¹xx−0 tan⁻¹x²/xin general. Letx /0 inX. Assumex∗yx∗z. Then tan⁻¹xx−y tan⁻¹xx−z. Since tan⁻¹ is a bijective mapping, we obtainxx−y xx−z, whencex /0 impliesxyxzandyz, that is, ifx /0, it is a left-injective element.

Sincex∗y ∈−π/2, π/2in that case, it follows thatx∗y π does not have a solution in such a case. Hencex∗X /X, that is,xis not ad-unit.

ByProposition 6.2, thed-algebra described inExample 6.3is ad-integral domain such that every element is not ad-unit. The following example shows that there is ad-algebra such that every non-zero element ofXis ad-unit, but not left-injective.

Example 6.4. LetX: 0,∞. Define a binary operation “∗” onXbyx∗y:x²x−y²for any x, y ∈ X. Then it is easy to show thatX,∗,0is ad-algebra. We claim that every non-zero elementxofXis ad-unit. Givenu∈X, if we takeyinXas follows:

y

⎧⎪

⎪⎨

⎪⎪

⎩ x²−√

u

x ify≤x, x²√

u

x otherwise,

6.1

then x∗y u, which proves thatxis ad-unit. We claim thatX is not left-injective, since 5∗6255∗4, but 6/4.

A non-empty subsetIof ad-algebraX,∗,0is said to be a left-ideal ofXif it satisfies the conditionD2. Every left-idealIof ad-algebraX,∗,0contains 0, since 0x∗x∈I∗X ⊆ Ifor somex∈I. Hence, a left-ideal ofXsatisfiesD0. Everyd-ideal of ad-algebraX,∗,0 is a left-ideal ofX, but the converse may not be true in general.

Example 6.5. LetX :{0, a, b, c}be ad-algebra which is not a BCK-algebra with the following table:

∗ 0 a b c 0 0 0 0 0 a a 0 1 a b b b 0 a c c c a 0

6.2

ThenJ :{0, a, c}is a left-ideal, but not ad-ideal ofX, sinceb∗c0∈Jandc∈J, butb /∈J.

Ad-algebraX,∗,0is said to be simple if its only left-ideals are{0}andX. Ad-algebra X,∗,0is said to bed-proper if for allx∈X,x∗Xis a left-ideal ofX.

Example 6.6. InExample 6.3, for anyx /0 inX, we havex∗X −π/2, π/2 X, that is, x∗Xis a left-ideal ofX. This shows thatX,∗,0isd-proper. It is not simple, sinceL∗X

−π/2, π/2⊆Lfor anyL⊆Xwith−π/2, π/2⊆L, that is,Lis a left-ideal ofX.

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Proposition 6.7. IfX,∗,0is a weakly associatived-algebra, then it isd-proper.

Proof. For anyx∈X, ifα∈x∗X∗X, thenα x∗y∗zfor somey, z∈X. SinceXis weakly associative,x∗y∗zx∗wfor somew∈X, that is,αx∗w∈x∗X, proving thatx∗Xis a left-ideal ofX.

Theorem 6.8. LetX,∗,0be ad-algebra. ThenXis simple andd-proper if and only if every non-zero element ofXis ad-unit.

Proof. SinceXisd-proper,x∗Xis a left-ideal ofXfor any non-zero elementxofX. Moreover, x /0 impliesx ∗ 0/0 and hence x ∗ X /{0}. By the simplicity of X,∗,0it follows that x ∗ XX, and thusxis ad-unit.

Assume that every non-zero element ofXis ad-unit. We claim thatXisd-proper. For anyx∈X, ifx0, then 0∗X {0}is a left-ideal ofX. Assumex /0. Sincexis ad-unit, we havex∗X Xand hencex∗X∗X X∗X ⊆Xx∗X, proving thatx∗Xis a left-ideal ofX. We claim thatXis simple. Assume thatLis a left-ideal ofXsuch thatL /{0}. If we let x /0 inL, thenXx∗X⊆L∗X⊆Lsincexis ad-unit. This proves thatXis simple.

ByTheorem 6.8, thed-algebraXdescribed inExample 6.4is simple andd-proper, but not left-injective.

Acknowledgment

The authors would like to express their great thanks to the referee’s careful reading and valuable suggestions.

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