aa r X i v : . [ m a t h . G R ] J un On inverse and right inverse ordered semigroups
A. Jamadar and K. Hansda
Abstract
A regular ordered semigroup S is called right inverse if every principal left ideal of S isgenerated by an R -unique ordered idempotent. Here we explore the theory of right inverseordered semigroups. We show that a regular ordered semigroup is right inverse if and only if anytwo right inverses of an element a ∈ S are R -related. Furthermore, different characterizations ofright Clifford, right group-like, group like ordered semigroups are done by right inverse orderedsemigroups. Thus a foundation of right inverse semigroups has been developed. Key Words and phrases: ordered regular, ordered inverse, ordered idempotent, completely regular,right inverse.
Right inverse semigroups are those, every element of which has unique right inverse. Thus naturallyit becomes generalization of inverse semigroups. Many extensive studies have been done on rightinverse semigroups by P.S. Venkatesan [13], G.L. Bailes [2] and some others. P.S. Venkatesan [13]studied these semigroups under the name of right unipotent semigroups. He showed that a semigroupis right inverse if and only if every right ideal of it generated by an idempotent.T. Saito [12], studied inverse semigroup by introducing simple ordered on it. Bhuniya andHansda [1] have deal with ordered semigroups in which any two inverses of an element are H -related. These ordered semigroups are the analouge of inverse semigroups. Hansda and Jamadar[9] named these ordered semigroups inverse ordered semigroups. They gave a detailed exposition onthe characterization of these ordered semigroups. Here we generalize such ordered semigroups intoright inverse ordered semigroups. This paper is inspired by the works done by P.S.Venkatesan [13],G.L.Bailes [2]. 1he presentation of this article is as follows: This section is followed by preliminaries. Section3 is devoted to the right inverse ordered semigroups. Here Clifford ordered semigroups have beencharacterized by right inverse semigroups. An ordered semigroup is a partiality ordered set ( S, · , ≤ ), and at the same time a semigroup ( S, · )such that for all a, b, x ∈ S a ≤ b implies xa ≤ xb and ax ≤ bx . It is denoted by ( S, · , ≤ ). Forevery subset H ⊆ S , denote ( H ] = { t ∈ S : t ≤ h, for some h ∈ H } . Throughout this article, unlessstated otherwise, S stands for an ordered semigroup and We assume that S does not contain the zeroelement.An equivalence relation ρ is called left (right) congruence if for a, b, c ∈ S aρb implies caρcb ( acρbc ).By a congruence we mean both left and right congruence. A congruence ρ is called semilattice con-gruence on S if for all a, b ∈ S, aρa and abρ ba . By a complete semilattice congruence we mean asemilattice congruence σ on S such that for a, b ∈ S, a ≤ b implies that aσab . The ordered semigroup S is called complete semilattice of subsemigroups of type τ if there exists a complete semilattice con-gruence ρ such that ( x ) ρ is a type τ subsemigroup of S . Let I be a nonempty subset of S . Then I iscalled a left(right) ideal of S , if SI ⊆ I ( IS ⊆ I ) and ( I ] ⊆ I . If I is both left and right ideal, then itis called an ideal of S . We call S a (left, right) simple ordered semigroup if it does not contain anyproper (left,right) ideal. For a ∈ S , the smallest (left, right) ideal of S that contains a is denoted by( L ( a ) , R ( a )) I ( a ). S is said to be regular (resp. Completely regular, right regular) ordered semigroup if for every a ∈ S, a ∈ ( aSa ]( a ∈ ( a Sa ] , a ∈ ( a S ]). Due to Kehayopulu [6] Green’s relations on a regularordered semigroup given as follows: a L b if L ( a ) = L ( b ), a R b if R ( a ) = R ( b ), a J b if I ( a ) = I ( b ), H = L ∩ R .This four relation L , R , J , and H are equivalence relation.A regular ordered semigroup S is said to be group-like (resp. left group-like) [1] ordered semi-group if for every a, b ∈ S, a ∈ ( Sb ] and b ∈ ( aS ](resp . a ∈ ( Sb ]). Right group like ordered semigroupcan be defined dually. A regular ordered semigroup S is called a right (left) Clifford [1] orderedsemigroup if for all a ∈ S, ( Sa ] ⊆ ( aS ] , (( aS ] ⊆ ( Sa ]). Every right (left) group like ordered semigroupis a right (left) Clifford ordered semigroup. An element b ∈ S is said to be an inverse of a ∈ S if a ≤ aba and b ≤ bab . The set of all inverses of an element a is denoted by V ≤ ( a ).2 heorem 2.1. [1] Let S be a regular ordered semigroup. Then the following statements are equivalent.1. S is right Clifford ordered semigroup;2. for all e ∈ E ≤ ( S ) , ( Se ] ⊆ ( eS ] ;3. for all a ∈ S , and e ∈ E ≤ ( S ) , there is x ∈ S such that ea ≤ ax ;4. for all a, b ∈ S , there is x ∈ S such that ba ≤ ax ;5. L ⊆ R on S . Lemma 2.2. [1] Let S be a right Clifford ordered semigroup. Then the following conditions hold in S . 1. a ∈ ( a Sa ] , for every a ∈ S ;2. ef ∈ ( f eSef ] , for every e, f ∈ E ≤ ( S ) . Theorem 2.3. [1] Let S be an ordered ordered semigroup. Then S is right (left) Clifford orderedsemigroup if and only if R ( L ) is the least complete semilattice congruence on S . Theorem 2.4. [1] Let S be a regular ordered semigroup. Then S is right (left) Clifford orderedsemigroup if and only if it is a complete semilattice of right (left) group like ordered semigroups. Let S be an ordered semigroup and ρ be an equivalence relation on S . In broad sense ρ -unique weshall mean the uniqueness in respect of the relation ρ . For example consider a subset T of S suchthat a, b are generators of T . Now if aρb we say that T is generated by ρ -unique element a . Definition 3.1.
A regular ordered semigroup S is called right inverse if every principal left ideal isgenerated by an R− unique ordered idempotent of S . We now present results on the role of ordered idempotents to characterize right inverse orderedsemigroups.
Theorem 3.2.
A regular ordered semigroup S is a right inverse if and only if for any two idempotents e, f ∈ E ≤ ( S ) , e L f implies e H f . Theorem 3.3.
Let S be a regular ordered semigroup. Then S is left (right) group like orderedsemigroup if and only if any two ordered idempotents are L ( R ) − related. orollary 3.4. Every right inverse left group like ordered semigroup is a group like ordered semi-group.
Theorem 3.5.
Let S be a regular ordered semigroup. Then any two inverses of an element are L− related if and only if ef ∈ ( eSf Se ] ; for some e, f ∈ E ≤ ( S ) . In the following theorem, we have shown that any two inverses of an element are R -related in aright inverse ordered semigroup. So in the broad sense they are R -unique. Theorem 3.6.
The following conditions are equivalent on a regular ordered semigroup S .1. S is right inverse;2. for a ∈ S and a ′ , a ′′ ∈ V ≤ ( a ) , a ′ R a ′′ ;3. for e, f ∈ E ≤ ( S ) , ef ∈ ( f SeSf ] ;4. ( eS ] ∩ ( f S ] = ( ef S ] ;5. for e ∈ E ≤ ( S ) and x ∈ ( Se ] implies x ′ ∈ ( eS ] , where x ∈ S and x ′ ∈ V ≤ ( x ) . Corollary 3.7.
Let S be a right inverse ordered semigroup. Then any two ordered idempotents e, f ∈ E ≤ ( S ) are H - commutative if and only if ( Se ] ∩ ( Sf ] = ( Sef ] . Example 3.8.
The ordered semigroup S = { a, e, f } defined by multiplication and order below. · a e fa a e fe a e ff a e f ′ ≤ ′ := { ( a, a ) , ( a, e ) , ( a, f ) , ( e, e ) , ( f, f ) } . From above table it is clear that a, e, f ∈ E ≤ ( S ) . Here ae ≤ aee = eaee = eaaee . So ae ∈ ( eSaSe ] . Also a ≤ aa implies that ea ≤ eaa = aeaa = aeeaa . So ea ∈ ( aSeSa ] . Similarly af ∈ ( f SaSf ] and f a ∈ ( aSf Sa ] . Also ef = f ef = f eef f that is ef ∈ ( f SeSf ] . Similarly it can beshown that f e ∈ ( eSf Se ] . Thus ( S, · , ≤ ) is a right inverse ordered semigroup. Theorem 3.9.
Let F be a semigroup. Then the ordered semigroup P f ( F ) of all subsets of F is aright inverse ordered semigroup if and only if F is a right inverse semigroup. Theorem 3.10.
Let S be a regular ordered semigroup. Then S is a right inverse ordered semigroupif and only if L e ⊆ ( R e ) ′ for any idempotent e in S . heorem 3.11. An ordered semigroup S is right Clifford if and only if S is right inverse and forevery a ∈ S , a ∈ ( a Sa ] . Theorem 3.12.
Let S be a right inverse ordered semigroup. If S is left Clifford then S is union ofgroup like ordered semigroups. In the following we show that in a right inverse ordered semigroup R is a congruence if and onlyif L = H . Theorem 3.13.
Let S be a right inverse ordered semigroup. The following are equivalent:1. R is a congruence on S ;2. L = H ;3. S is a complete semilattice of right group like ordered semigroups. Our paper ends up with the corollary that follows from Theorem 3 .
13 and Theorem 3 .
12 andwhich gives a characterization on right inverse semigroup to become a completely regular orderedsemigroup.
Corollary 3.14.
Let S be a right inverse and left regular ordered semigroup. Then following condi-tions are equivalent.1. R is a congruence on S ;2. L = H ;3. S is a complete semilattice of right group like ordered semigroups;4. S is completely regular. References [1] A. K. Bhuniya and K. Hansda, Complete semilattice of ordered semigroups, Communicated.[2] G. L. Bailes, Right inverse Semigroups,
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