Fully fuzzy prime semigroups

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M. SHABIR

Received 25 March 2004

We characterize those semigroups for which each fuzzy ideal is prime. We also character-ize those semigroups for which each fuzzy right ideal is prime.

1. Introduction and preliminaries

The fundamental concept of a fuzzy set, introduced by Zadeh in his classic paper [5] of 1965, has been applied by many authors to generalize some of the basic notions of algebra. In this note we characterize the semigroups for which each fuzzy ideal is prime, also the semigroups for which each fuzzy right ideal is prime.

InSection 2, we prove that a semigroup is fully fuzzy prime if and only if it is semisim-ple and its set of fuzzy ideals is totally ordered. InSection 3, we define the fuzzy prime right ideals of a semigroup and we prove that if the set of all fuzzy right ideals ofS is totally ordered, thenSis right weakly regular if and only if every fuzzy right ideal ofSis a fuzzy prime right ideal.

By asemigroup(S,·), we mean a nonempty setStogether with an associative binary operation “·”. A semigroupSis calledcommutativeif “·” is commutative, that is,a·b= b·afor alla,b∈S. A semigroupSis called amonoidif it has an identity element with respect to “·”. IfShas no identity element, then it is easy to adjoin an identity element 1 to the set by defining 1·s=s·1=s, for allsinS. We will use the notationS1 _{with the} following meaning:

S1₌   

S ifShas an identity element,

S∪ {1} otherwise. (1.1)

A semigroupSis calledregular, if for eachainSthere existsxinSsuch thataxa=a. Ifais an element of a semigroupS, we say thatxis aninverseofaifaxa=aandxax=x. A semigroupSis called aninverse semigroupif each of its elements has a unique inverse.

By aleft (right) idealofS, we mean a nonempty subsetAofSsuch thatSA⊆A(AS⊆ A). By atwo-sided idealor simply anideal, we mean a nonempty subset ofS, which is both a left and a right ideal ofS. An idealPof a semigroupSis calledprime (semiprime)

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idealofSif for idealsI,JofS,IJ⊆P(I2_⊆_P_{) implies}_I_⊆_P_or_J_⊆_P₍_I_⊆_P_);_P_{is called} irreducibleif for idealsI,JofS,I∩J=PimpliesI=PorJ=P.

A function f from a nonempty setAto the unit interval [0, 1] is called afuzzy subset ofA. A fuzzy subsetλ:A→[0, 1] isnonemptyifλis not the constant map which assumes the value 0. For fuzzy subsetsλandµofA,λ≤µmeans that for alla∈A,λ(a)≤µ(a). The symbolsλ∧µandλ∨µdenote the following fuzzy subsets ofA:

(λ∧µ)(a)=λ(a)∧µ(a) (a∈A),

(λ∨µ)(a)=λ(a)∨µ(a) (a∈A). (1.2)

By the productλ◦µof two fuzzy subsetsλ,µof a semigroupS, we mean the following fuzzy subset:

(λ◦µ)(x)=

  ∨x=yz

_λ

(y)∧µ(z) ifxis expressible asx=yz

0 otherwise. (1.3)

The operation◦is associative.

A nonempty fuzzy subset f of Sis called afuzzy left (right) idealof Sif fS◦f ≤ f,

(f ◦fS≤ f), where fS:S→[0, 1] maps every element ofSto 1. A nonempty fuzzy subset f ofSis called afuzzy two-sided idealofSif it is both a fuzzy left and a fuzzy right ideal ofS.

The following lemmas are taken from [2].

Lemma1.1. A fuzzy subsetf of a semigroupSis a fuzzy left ideal ofSif and only if f(xy)≥ f(y)for allx,y∈S.

Lemma1.2. Let f,g, andhbe fuzzy subsets ofS, then (i) f ∨(g∧h)=(f ∨g)∧(f∨h),

(ii) f ∧(g∨h)=(f ∧g)∨(f∧h).

Lemma1.3. Let f,g, andhbe fuzzy subsets ofS, then

(i) f ◦(g∨h)=(f◦g)∨(f ◦h),(g∨h)◦f =(g◦f)∨(h◦f), (ii) f ◦(g∧h)≤(f◦g)∧(f ◦h),(g∧h)◦f ≤(g◦f)∧(h◦f).

Lemma1.4. Let f,g, andhbe fuzzy subsets ofS. If f ≤g then f ◦h≤g◦handh◦f ≤ h◦g.

Lemma1.5. If f is a nonempty fuzzy subset ofS, then fS◦f (f ◦fS)is a fuzzy left (right)

ideal ofS.

Lemma1.6. If f is a fuzzy right (left) ideal ofS, then f ∨(fS◦f) (f ∨(f◦fS))is a fuzzy two-sided ideal ofS.

Lemma1.7. Ifλandµare fuzzy ideals ofS, thenλ∧µandλ∨µare fuzzy ideals ofS.

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2. Fully fuzzy prime semigroups

Definition 2.1. A semigroupSis calledsemisimpleif each of its ideals is idempotent. Definition 2.2. A fuzzy idealξof a semigroupSis calledfuzzy prime idealofSif for any fuzzy idealsλ,µofS,λ◦µ≤ξimpliesλ≤ξorµ≤ξ,ξis calledfuzzy irreducibleif for any fuzzy idealsλ,µofS,λ∧µ=ξimpliesλ=ξorµ=ξ. A fuzzy idealξof a semigroupSis calledfuzzy semiprime idealofSif for any fuzzy idealλofS,λ2_≤_ξ_implies_λ_≤_ξ_.

Proposition2.3. A fuzzy ideal of a semigroupSis fuzzy prime ideal if and only if it is fuzzy semiprime and fuzzy irreducible.

Proof. Letλbe a fuzzy prime ideal of a semigroupS, thenλis fuzzy semiprime ideal. Let µ,νbe fuzzy ideals ofSsuch thatµ∧ν=λ. Sinceµ◦ν≤µ∧ν=λandλis fuzzy prime ideal ofS, we haveµ≤λorν≤λ. Sinceλ≤µandλ≤ν, we haveλ=µorλ=ν. Henceλ is irreducible.

Conversely, assume that λis a fuzzy ideal ofS, which is both fuzzy semiprime and fuzzy irreducible. Ifµ,νare fuzzy ideals ofSsuch thatµ◦ν≤λ, then (µ∧ν)2_≤_µ_◦_ν_≤_λ_. Sinceλ is fuzzy semiprime, we haveµ∧ν≤λ. Thus (µ∧ν)∨λ=λ. This implies that (µ∨λ)∧(ν∨λ)=λ. Sinceλis fuzzy irreducible, we haveµ∨λ=λorν∨λ=λ. That is

eitherµ≤λ, orν≤λ.

Definition 2.4. A fuzzy idealλof a semigroupSis called idempotent ifλ◦λ=λ.

Definition 2.5. A semigroupSis called fully fuzzy prime (semiprime) if each of its fuzzy ideal is prime (semiprime).

Theorem2.6. The following assertions on a semigroupSare equivalent: (1)Sis semisimple,

(2)each fuzzy ideal ofSis idempotent,

(3)for each pair of fuzzy idealsλ,µofS,λ∧µ=λ◦µ,

(4)the set of all fuzzy ideals ofS(ordered by inclusion) is a distributive lattice under the union and product of fuzzy ideals,

(5)each proper fuzzy ideal ofSis the intersection of fuzzy prime ideals, (6)each fuzzy ideal ofSis fuzzy semiprime ideal.

Proof. (1)⇔(2)⇔(3). This is due to Kuroki [4]. (1)⇔(4)⇔(5). This is proved in [3].

(1)⇒(6). Letλbe a fuzzy ideal ofS. Letδbe a fuzzy ideal ofSsuch thatδ2_≤_λ_{. Now by} (1),δ2₌_δ_{, so}_δ_≤_λ_{. Hence,}_λ_{is fuzzy semiprime ideal.}

(6)⇒(1). Letλbe any fuzzy ideal ofS. Thenλ2_{is also a fuzzy ideal of}_S_{. Since}_λ2_≤_λ2_, by (6),λ≤λ2_{. We have on the other hand}_λ2_≤_λ_{, so}_λ₌_λ2_.

Theorem2.7. LetSbe a semisimple semigroup. For a fuzzy idealξofS, the following con-ditions are equivalent:

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Proof. Assume thatξis a fuzzy prime ideal ofS. We show thatξis fuzzy irreducible, that is, for fuzzy idealsλ,µofS,λ∧µ=ξimpliesλ=ξ orµ=ξ. Sinceλ∧µ=ξ, it follows thatξ≤λandξ≤µ. ByTheorem 2.6,λ◦µ=λ∧µ. Hence,ξ=λ∧µ=λ◦µ. Sinceξis a fuzzy prime ideal, we have eitherλ≤ξorµ≤ξ. Soξ=λorξ=µ.

Conversely, assume thatξis a fuzzy irreducible ideal. We show thatξis a fuzzy prime ideal. Suppose that there exist fuzzy idealsλandµsuch thatλ◦µ≤ξ. ByTheorem 2.6,λ◦ µ=λ∧µ. Thus, (λ∧µ)∨ξ=ξimplies (λ∨ξ)∧(µ∨ξ)=ξ. Sinceξis fuzzy irreducible, it follows that eitherλ∨ξ=ξ orµ∨ξ=ξ. Then we haveλ≤ξ orµ≤ξ. Hence,ξ is a

fuzzy prime ideal.

Theorem2.8. A semigroupSis fully fuzzy prime if and only ifSis semisimple and the set of fuzzy ideals ofSis totally ordered.

Proof. SupposeSis a fully fuzzy prime semigroup,λis a fuzzy ideal ofS, thenλ2_{is also a} fuzzy ideal ofS. Sinceλ2_≤_λ2_{, we have}_λ_≤_λ2_{. On the other hand,}_λ2_≤_λ_{. Hence}_λ2₌_λ_. Thus, every fuzzy ideal ofSis idempotent, soSis semisimple. Letλandµbe fuzzy ideals ofS, thenλ∧µis a fuzzy ideal and so fuzzy prime ideal ofS. Sinceλ◦µ≤λ∧µ, then eitherλ≤λ∧µorµ≤λ∧µ. Hence,λ≤µorµ≤λ.

Conversely, assume thatSis a semisimple semigroup and the set of fuzzy ideals ofS is totally ordered. Letλ,µ, andξ be fuzzy ideals ofSsuch thatλ◦µ≤ξ. Since the set of fuzzy ideals ofSis totally ordered, so we haveλ≤µorµ≤λ. Assume thatλ≤µ. Now

λ=λ2_≤_λ_◦_µ_≤_ξ_{. Thus,}_ξ_{is a fuzzy prime ideal of}_S_.

Corollary2.9. A commutative semigroupSis fully fuzzy prime if and only ifSis an inverse semigroup and the set of fuzzy ideals ofSis totally ordered.

Proof. Since each commutative semisimple semigroup is an inverse semigroup, the

corol-lary follows from the above theorem.

3. Fuzzy prime right ideals

Throughout this section,Swill denote a monoid, that is, a semigroup with an identity 1, which also contains a two-sided zero.

Definition 3.1. A fuzzy right idealξof a semigroupSis calledfuzzy prime right idealof Sif for any fuzzy right idealsλ,µofS,λ◦µ≤ξimpliesλ≤ξ orµ≤ξ;ξis calledfuzzy irreducibleif for any fuzzy right idealsλ,µofS,λ∧µ=ξimpliesλ=ξ orµ=ξ. A fuzzy right idealξof a semigroupSis calledfuzzy semiprime right idealofSif for any fuzzy right idealλofS,λ2_≤_ξ_implies_λ_≤_ξ_.

Definition 3.2. A semigroupSis calledright weakly regularif for eachx∈S,x∈(xS)2_(cf. [1]).

Lemma3.3. IfSis a semigroup, then the intersection of fuzzy prime right ideals ofSis a fuzzy semiprime right ideal.

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Proof. Letξ be a fuzzy semiprime irreducible right ideal ofS. Let λ,µbe fuzzy right ideals ofS, such thatλ◦µ≤ξ. Now we show thatµ≤ fS◦µ. Letx∈S. Then (fS◦µ)(x)=

∨x=yz{fS(y)∧µ(z)} = ∨x=yz{µ(z)}because fS(y)=1≥µ(x), as we havex=1x.

Hence,µ∨fS◦µ=fS◦µis a fuzzy ideal ofS. Alsoλ∧fS◦µis a fuzzy right ideal ofS. Now (λ∧fS◦µ)2≤λ◦(fS◦µ)=(λ◦fS)◦µ≤λ◦µ≤ξ.

Sinceξis fuzzy semiprime right ideal,λ∧fS◦µ≤ξ. Thus (λ∧fS◦µ)∨ξ=ξimplies

(λ∨ξ)∧(fS◦µ∨ξ)=ξ.

Sinceξ is fuzzy irreducible, we have (λ∨ξ)=ξor (fS◦µ∨ξ)=ξ that is eitherλ≤ξ

orfS◦µ≤ξimplies eitherλ≤ξorµ≤ξ. Hence,ξis a fuzzy prime right ideal ofS.

Theorem3.5. The following assertions on a semigroupSare equivalent. (1)Sis right weakly regular,

(2)each fuzzy right ideal ofSis idempotent,

(3)for each fuzzy idealµand for each fuzzy right idealλofS,λ∧µ=λ◦µ, (4)each fuzzy right ideal ofSis a fuzzy semiprime right ideal,

(5)each fuzzy right ideal is the intersection of those fuzzy prime right ideals ofSwhich contain it.

Proof. (1)⇔(2)⇔(3). Compare to [1].

(1)⇒(4). Letξbe a fuzzy right ideal ofS. Letλ2_≤_ξ_{, where}_λ_{is a fuzzy right ideal of}_S_. By (2)λ2₌_λ_{, so}_λ_≤_ξ_{. Thus}_ξ_{is a fuzzy semiprime right ideal of}_S_.

(4)⇒(5). First, we show that each fuzzy right ideal ofSis contained in a fuzzy irre-ducible right ideal. Letλbe a fuzzy right ideal ofSwithλ(a)=α, whereais any element ofSandα∈[0, 1]. LetX= {µ:µis a fuzzy right ideal ofS,µ(a)=α, andλ≤µ}. Then X= ∅sinceλ∈X. Defineξ= ∨X, thenξis a maximal fuzzy right ideal ofSsatisfying ξ(a)=α. We now show thatξis a fuzzy irreducible right ideal ofS. Supposeξ=δ1∧δ2, whereδ1andδ2are fuzzy right ideals ofS. This implies thatξ≤δ1andξ≤δ2. We claim that eitherξ=δ1orξ=δ2. Suppose, on the contrary,ξ=δ1andξ=δ2. Sinceξis maxi-mal with respect to the property thatξ(a)=αand sinceξδ1andξδ2, it follows that δ1(a)=αandδ2(a)=α. Henceα=ξ(a)=(δ1∧δ2)(a)= {δ1(a)∧δ2(a)} =α, which is impossible. Hence,ξ=δ1orξ=δ2. This proves thatξis a fuzzy irreducible ideal.

Now letλ be a fuzzy right ideal ofSand let {λα:α∈Ω}be the family of fuzzy

ir-reducible right ideals ofSwhich containλ. Obviously,λ≤ ∧α∈Ωλα. We now prove that

∧α∈Ωλα≤λ. Letabe any element ofS. Then there exists a fuzzy irreducible right ideal, sayλβ, such thatλ≤λβandλ(a)=λβ(a). Thusλβ∈ {λα:α∈Ω}. Hence∧α∈Ωλα≤λβ, so

∧α∈Ωλα(a)≤λβ(a)=λ(a). This implies that∧α∈Ωλα≤λ. Hence, each fuzzy right ideal of

Scan be written as the intersection of fuzzy irreducible right ideals ofSwhich contain it. By (4), each fuzzy right ideal is semiprime, so byLemma 3.4, each fuzzy right ideal ofSis the intersection of fuzzy prime right ideals ofSwhich contain it.

(5)⇒(1). Letλbe a fuzzy right ideal ofS, thenλ2_{is also a fuzzy right ideal of}_S_{. By} hy-pothesis,λ2 _{is an intersection of fuzzy prime right ideals and so is a fuzzy semiprime} right ideal of S. As λ2_≤_λ2_{, so} _λ_≤_λ2_{. On the other hand,} _λ2_≤_λ_{. Hence} _λ2₌_λ_{. As}

(2) is equivalent to (1), thusSis right weakly regular.

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Proof. SupposeSis a semigroup in which each fuzzy right ideal is prime and letλbe a fuzzy right ideal ofS. Thenλ2_{is also a fuzzy right ideal of}_S_{. Since}_λ2_≤_λ2_{, we have}_λ_≤_λ2_. On the other handλ2_≤_λ_{. Hence}_λ2₌_λ_{. Thus, every fuzzy right ideal of}_S_{is idempotent,} soSis right weakly regular. Letλandµbe any fuzzy ideals ofS. Sinceλ◦µ≤λ∧µ. As λ∧µis a fuzzy ideal ofS, then it is, a fuzzy prime ideal. Thus eitherλ≤λ∧µorµ≤λ∧µ,

and therefore eitherλ≤µorµ≤λ.

Proposition3.7. IfSis a right weakly regular semigroup and the set of all fuzzy right ideals ofSis totally ordered, then every fuzzy right ideal ofSis a fuzzy prime right ideal.

Proof. Letλ,µ, andξbe fuzzy right ideals ofSsuch thatλ◦µ≤ξ. Since the set of all fuzzy right ideals ofSis totally ordered, we haveλ≤µorµ≤λ. Ifλ≤µ, thenλ=λ2_≤_λ_◦_µ_≤_ξ_. Ifµ≤λ, thenµ=µ2_≤_λ_◦_µ_≤_ξ_{. Thus}_ξ_{is a fuzzy prime right ideal.}

Theorem3.8. If the set of fuzzy right ideals ofSis totally ordered, then the following are equivalent.

(1)Sis right weakly regular,

(2)every fuzzy right ideal ofSis a fuzzy prime right ideal.

References

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[3] J. Ahsan, K. Saifullah, and M. F. Khan,Semigroups characterized by their fuzzy ideals, Mohu Xitong yu Shuxue9(1995), no. 1, 29–32.

[4] N. Kuroki,On fuzzy semigroups, Inform. Sci.53(1991), no. 3, 203–236. [5] L. A. Zadeh,Fuzzy sets, Inform. and Control8(1965), no. 3, 338–353.

10.1155/IJMMS.2005.163