aa r X i v : . [ m a t h . G R ] J un Idempotent ordered semigroup
K. Hansda
Department of Mathematics, Visva-Bharati University,Santiniketan, Bolpur - 731235, West Bengal, [email protected]
June 27, 2017
Abstract
An element e of an ordered semigroup ( S, · , ≤ ) is called an ordered idempotent if e ≤ e . Wecall an ordered semigroup S idempotent ordered semigroup if every element of S is an orderedidempotent. Every idempotent semigroup is a complete semilattice of rectangular idempotentsemigroups and in this way we arrive to many other important classes of idempotent orderedsemigroups. Idempotents play an important role in different major subclasses of a regular semigroup. A regularsemigroup S is called orthodox if the set of all idempotents E ( S ) forms a subsemigroup, and S is aband if S = E ( S ).T. Saito studied systematically the influence of order on idempotent semigroup [8]. In [1],Bhuniya and Hansda introduced the notion of ordered idempotents and studied different classes ofregular ordered semigroups, such as, completely regular, Clifford and left Clifford ordered semigroupsby their ordered idempotents. The purpose of this paper to study ordered semigroups in whichevery element is an ordered idempotent. Complete semilattice decomposition of these semigroupsautomatically suggests the looks of rectangular idempotent semigroups and in this way we arrive tomany other important classes of idempotent ordered semigroups.1he presentation of the article is as follows. This section is followed by preliminaries. InSection 3, is devoted to the idempotent ordered semigroups and characterizations of different type ofidempotent ordered semigroups. An ordered semigroup is a partially ordered set ( S, ≤ ), and at the same time a semigroup ( S, · ) suchthat for all a, b, c ∈ S ; a ≤ b implies ca ≤ cb and ac ≤ bc . It is denoted by ( S, · , ≤ ). Throughout thisarticle, unless stated otherwise, S stands for an ordered semigroup. For every subset H ⊆ S , denote( H ] = { t ∈ S : t ≤ h, for some h ∈ H } . Let I be a nonempty subset of an ordered semigroup S . I is a left (right) ideal of S , if SI ⊆ I ( IS ⊆ I ) and ( I ] = I . I is an ideal of S if I is both a left anda right ideal of S . An (left, right) ideal I of S is proper if I = S . S is left (right) simple if it doesnot contain proper left (right) ideals. S is called t-simple if it is both left and right simple and S iscalled simple if it does not contain any proper ideals.Kehayopulu [2] defined Green’s relations on a regular ordered semigroup S as follows: a L b if ( S a ] = ( S b ] , a R b if ( aS ] = ( bS ] , a J b if ( S aS ] = ( S bS ] , and H = L ∩ R . These four relations L , R , J and H are equivalence relations.An equivalence relation ρ on S is called left (right) congruence if for every a, b, c ∈ S ; a ρ b implies ca ρ cb ( ac ρ bc ). By a congruence we mean both left and right congruence. A congruence ρ is calleda semilattice congruence on S if for all a, b ∈ S, a ρ a and ab ρ ba . By a complete semilatticecongruence we mean a semilattice congruence σ on S such that for a, b ∈ S, a ≤ b implies that aσab .Equivalently: There exists a semilattice Y and a family of subsemigroups { S } α ∈ Y of type τ of S suchthat:1. S α ∩ S β = φ for any α, β ∈ Y with α = β, S = S α ∈ Y S α , S α S β ⊆ S α β for any α, β ∈ Y, S β ∩ ( S α ] = φ implies β (cid:22) α, where (cid:22) is the order of the semilattice Y defined by (cid:22) := { ( α, β ) | α = α β ( β α ) } [7].An ordered semigroup S is called left regular if for every a ∈ S, a ∈ ( Sa ]. An element e of ( S, · , ≤ ) is called an ordered idempotent [1] if e ≤ e . As an immediate example of idempotentordered semigroups, we can consider ( N , · , ≤ ) which is an idempotent ordered semigroup but ( N , · )2s not an idempotent semigroup. An ordered semigroup S is called H− commutative if for every a, b ∈ S, ab ∈ ( bSa ].If F is a semigroup, then the set P f ( F ) of all finite subsets of F is a semilattice ordered semigroupwith respect to the product ′ · ′ and partial order relation ′ ≤ ′ given by: for A, B ∈ P f ( F ), A · B = { ab | a ∈ A, b ∈ B } and A ≤ B if and only if A ⊆ B. We have discussed the elementary properties of ordered idempotents. In this section we characterizeordered semigroups of which every element is an ordered idempotent. Here we show that theseordered semigroups are analogous to bands.We now make a natural analogy between band and idempotent ordered semigroup.
Theorem 3.1.
Let B be a semigroup. Then P f ( B ) is idempotent ordered semigroup if and only if B is a band.Proof. Let B be a band and U ∈ P f ( B ). Choose x ∈ U . Then x ∈ U implies x ∈ U . Then U ⊆ U . So P f ( B ) is idempotent ordered semigroup.Conversely, assume that B be a semigroup such that P f ( B ) is an idempotent ordered semigroup.Take y ∈ B . Then Y = { y } ∈ P f ( B ). Thus Y ⊆ Y , which implies y = y . Hence B is a band. Proposition 3.2.
Let B be a band, S be an idempotent ordered semigroup and f : B −→ S be asemigroup homomorphism. Then there is an ordered semigroup homomorphism φ : P f ( B ) −→ S such that the following diagram is commutative: B SP f ( B ) fl φ where l : B −→ P f ( B ) is given by l ( x ) = { x } .Proof. Define φ : P f ( F ) −→ S by: for A ∈ P f ( F ), φ ( A ) = ∨ a ∈ A f ( a ). Then for every A, B ∈ P f ( F ) , φ ( AB ) = ∨ a ∈ A,b ∈ B f ( ab ) = ∨ a ∈ A,b ∈ B f ( a ) f ( b ) = ( ∨ a ∈ A f ( a ))( ∨ b ∈ B f ( b )) = φ ( A ) φ ( B ) , andif A ≤ B , then φ ( A ) = ∨ a ∈ A f ( a ) ≤ ∨ b ∈ B f ( b ) = φ ( B ) shows that φ is an ordered semigrouphomomorphism. Also φ ◦ l = f . 3 emma 3.3. In an idempotent ordered semigroup
S, a m ≤ a n f or every a ∈ S and m, n ∈ N with m ≤ n . Every idempotent ordered semigroup S is completely regular and hence J is the least completesemilattice congruence on S , by Lemma ?? . In an idempotent ordered semigroup S , the Green’srelation J can equivalently be expressed as: for a, b ∈ S , a J b if there are x, y, u, v ∈ S such that a ≤ axbya and b ≤ buavb. Now we characterize the
J − class in an idempotent ordered semigroup.
Definition 3.4.
An idempotent ordered semigroup S is called rectangular if for all a, b ∈ S , thereare x, y ∈ S such that a ≤ axbya . Example 3.5. ( N , · , ≤ ) is a rectangular idempotent ordered semigroup,whereas if we define a o b = min { a, b } f or all a, b ∈ N then ( N , ◦ , ≤ ) is an idempotent ordered semigroup but not rectangular. Also we have the following equivalent conditions.
Lemma 3.6.
Let S be an idempotent ordered semigroup. Then the following conditions are equivalent:1. S is rectangular;2. for all a, b ∈ S , there is x ∈ S such that a ≤ axbxa ;3. for all a, b, c ∈ S there is x ∈ S such that ac ≤ axbxc .Proof. (1) ⇒ (3): Let a, b, c ∈ S . Then there are x, y ∈ S such that a ≤ axbya . This implies that ac ≤ axbyac ≤ ax ( bya )( bya ) c ≤ ( axbyab )( axbyab ) yac ≤ a ( axbyabya ) b ( axbyabya ) c ≤ atbtc, where t = axbyabya ∈ S .(3) ⇒ (2): Let a, b ∈ S . Then there is x ∈ S such that a ≤ axbxa . Then a ≤ a implies that a ≤ axbxa .(2) ⇒ (1): This follows directly.As we can expect, we show that the equivalence classes in an idempotent ordered semigroup S determined by J are rectangular. Theorem 3.7.
Every idempotent ordered semigroup is a complete semilattice of rectangular idempo-tent ordered semigroups. roof. Let S be an idempotent ordered semigroup. Then J is the least complete semilattice congru-ence on S . Now consider a J -class ( c ) J for some c ∈ S . Since J is a complete semilattice congruence,( c ) J is a subsemigroup of S . Let a, b ∈ ( c ) J . Then there is x ∈ S such that a ≤ axbxa , which impliesthat a ≤ a ( axb ) b ( bxa ) a , that is, a ≤ aubva where u = axb and v = bxa . Also the completeness of J implies that ( a ) J = ( a xbxa ) J = ( axb ) J = ( bxa ) J , and u, v ∈ ( c ) J . Thus ( c ) J is a rectangularidempotent ordered semigroup. Definition 3.8.
An idempotent ordered semigroup S is called left zero if for every a, b ∈ S , thereexists x ∈ S such that a ≤ axb . An idempotent ordered semigroup S is called right zero if for every a, b ∈ S , there is x ∈ S suchthat a ≤ bxa .Now it is evident that every left zero and right zero idempotent ordered semigroup is rectangular. Proposition 3.9.
An idempotent ordered semigroup is left zero if and only if it is left simple.Proof.
First suppose that S is a left zero idempotent ordered semigroup and a ∈ S . Then for any b ∈ S , there is x ∈ S such that b ≤ bxa , which shows that b ∈ ( Sa ]. Thus S = ( Sa ] and hence S isleft simple.Conversely, assume that S is left simple. So for every a, b ∈ S , there is s ∈ S such that a ≤ sb .Then a ≤ a gives that a ≤ asb . Thus S is a left zero idempotent ordered semigroup. Lemma 3.10.
In an idempotent ordered semigroup S , the following conditions are equivalent:1. For all a, b ∈ S , there is x ∈ S such that ab ≤ abxba ;2. For all a, b ∈ S , there is x ∈ S such that ab ≤ axbxa ;3. For all a, b ∈ S , there is x, y ∈ S such that ab ≤ axbya .Proof. (1) ⇒ (3): This follows directly.(3) ⇒ (2): This is similar to the Lemma 3.6.(2) ⇒ (1): Let a, b ∈ S . Then there is x ∈ S such that bab ≤ baxbxba . Now since ab ≤ ababab, we have ab ≤ ab ( abaxbx ) ba .We now introduce the notion of left regularity in an idempotent ordered semigroup. Definition 3.11.
An idempotent ordered semigroup S is called left regular if for every a, b ∈ S thereis x ∈ S such that ab ≤ axbxa . heorem 3.12. An idempotent ordered semigroup S is left regular if and only if L = J is the leastcomplete semilattice congruence on S .Proof. First we assume that S is left regular. Let a, b ∈ S be such that a J b . Then there are u, v, x, y ∈ S such that a ≤ ubv and b ≤ xay . Since S is left regular, there are s, t ∈ S suchthat bv ≤ bsvsb and ay ≤ atyta . Then a ≤ ubsvsb and b ≤ xatyta ; which shows that a L b . Thus J ⊆ L . Again
L ⊆ J on every ordered semigroup and hence L = J . Since every idempotent orderedsemigroup is completely regular, it follows that L is the least complete semilattice congruence on S ,by Lemma ?? Conversely, let L is the least complete semilattice congruence on S . Consider a, b ∈ S . Then ab L ba implies that ab ≤ xba for some x ∈ S . This implies that ab ≤ abab ≤ abxba. Thus S is a left regular idempotent ordered semigroup, by Lemma 3.10.In the following result a left regular idempotent ordered semigroup has been characterized byleft zero idempotent ordered semigroup. Theorem 3.13.
Let S be an idempotent ordered semigroup. Then the following conditions areequivalent:1. S is left regular;2. S is a complete semilattice of left zero idempotent ordered semigroups ;3. S is a semilattice of left zero idempotent ordered semigroups.Proof. (1) ⇒ (2): In view of Theorem 3.12, it is sufficient to show that each L -class is a left zeroidempotent ordered semigroup. Let L be an L -class and a, b ∈ L . Then L is a subsemigroup,since L is a semilattice congruence. Since a L b there is x ∈ S such that a ≤ xb . This implies that a ≤ a ≤ a xb ≤ a xb ≤ aub, where u = axb .By the completeness of L , a ≤ xb implies that ( a ) L = ( axb ) L , and hence u ∈ L . Thus S is leftzero idempotent ordered semigroup.(2) ⇒ (3): Trivial.(3) ⇒ (1): Let ρ be a semilattice congruence on S such that each ρ -class is a left zero idempotentordered semigroup. Consider a, b ∈ S . Then ab, ba ∈ ( ab ) ρ shows that there is s ∈ ( ab ) ρ such that ab ≤ absba ≤ absbsba ≤ a ( bsb ) b ( bsb ) a . Hence S is left regular.6he following theorem gives several alternative characterizations of an H− commutative idem-potent ordered semigroup. Lemma 3.14.
Let S be an idempotent ordered semigroup. Then the following conditions are equiv-alent:1. S is H -commutative;2. for all a, b ∈ S, ab ∈ ( baS ] ∩ ( Sba ]; S is a complete semilattice of t-simple idempotent ordered semigroups;4. S is a semilattice of t-simple idempotent ordered semigroups.Proof. (1) ⇒ (2): Consider a, b ∈ S . Since S is H− commutative, there is u ∈ S such that ab ≤ bua . Also for u, a ∈ S, ua ≤ asu for some s ∈ S . Thus ab ≤ basu , which showsthat ab ∈ ( baS ]. Similarly ab ∈ ( Sba ]. Hence ab ∈ ( baS ] ∩ ( Sba ].(2) ⇒ (3): Suppose that J be an J -class in S and a, b ∈ J . Since J is rectangular there is x ∈ J such that a ≤ axbxa . Also by the given condition (2) there is u ∈ J such that bxa ≤ xaub .So a ≤ ax aub ≤ vb where v = ax au . Since J is a complete semilattice congruence on S, ( a ) J =( a x aub ) J = ( ax au ) J = ( v ) J . So v ∈ J . This shows that J is left simple. Similarly it can beshown that J is also right simple. Thus S is a complete semilattice of t-simple idempotent orderedsemigroups.(3) ⇒ (4): This follows trivially.(4) ⇒ (1): Let S be the semilattice Y of t-simple idempotent ordered semigroups { S α } α ∈ Y and ρ be the corresponding semilattice congruence on S . Then there are α, β ∈ Y such that a ∈ S α and b ∈ S β . Then ba, ab ∈ S αβ . Since S αβ is t-simple, ab ≤ xba for some x ∈ S αβ . Now for x, ba ∈ S αβ thereis y ∈ S αβ such that x ≤ bay . This finally gives ab ≤ bta , where t = ayb . Definition 3.15.
An idempotent ordered semigroup ( S, ., ≤ ) is called weakly commutative if for any a, b ∈ S there exists u ∈ S such that ab ≤ bua . Theorem 3.16.
For an idempotent ordered semigroup S , the followings are equivalent:1. S is weakly commutative;2. for any a, b ∈ S, ab ∈ ( baS ] ∩ ( sba ]; S is complete semilattice of left and right simple idempotent ordered semigroups .Proof. (1) ⇒ (2): Let a, b ∈ S . Then ∃ u ∈ S s.t ab ≤ bua for u, a ∈ S, there exists z ∈ S such that ua ≤ azu . Thus ab ≤ bua ≤ baza f or za ∈ S . So ab ≤ ( baS ]. Similarly ab ∈ ( Sba ].Hence ab ∈ ( baS ] ∩ ( sba ]. 72) ⇒ (3): Since S is an idempotent ordered semigroup, we have by theorem[] ρ is completesemilattice congruence. We now have to show that, for each z ∈ S, J = ( z ) ρ is left and right simple.For this let us choose a, b ∈ J . Then there exists x, y ∈ S such that a ≤ axbya . So from the givencondition bya ∈ ( syab ] and therefore there is s ∈ S such that bya ≤ s yab . Therefore a ≤ axs yab .Now since ρ is complete semilattice congruence on S , we have ( a ) ρ = ( a xs yab ) ρ = ( axs yab ) ρ =( axs ya ) ρ . Thus a ≤ ub, where u = axs ya ∈ J . Hence J is left simple and similarly it is rightsimple.(3) ⇒ (1): Let S is complete semilattice Y of left and right simple idempotent ordered semi-groups { S α } α ∈ Y . Thus S = { S α } α ∈ Y . Take a, b ∈ S . Then there are α, β ∈ Y such that a ∈ S α and b ∈ S β . Thus ab ∈ S αβ . So ab, ba ≤ S αβ . Then there are u, v ∈ S αβ such that ab ≤ uba and ab ≤ bav implies ab ≤ ab ≤ bta, where t = avub . Hence S is weakly commutative. Thiscompletes the proof. Definition 3.17.
An idempotent ordered semigroup ( S, · , ≤ ) is called normal if for any a, b, c ∈ S, ∃ x ∈ S such that abca ≤ acxba . Theorem 3.18.
For an idempotent ordered semigroup S , the followings are equivalent:1. S is normal;2. aSb is weakly commutative, for any a, b ∈ S ;3. aSa is weakly commutative, for any a ∈ S .Proof. (1) ⇒ (2): Consider axb, ayb ∈ aSb f or x, y ∈ S . As S is normal, ∃ u, v ∈ S such that( axb )( ayb ) ≤ ( axb )( ayb )( axb )( ayb ) ≤ aybuxba xbayb, f or xba, yb ∈ S ≤ ( ayb ) uxb ( bay ) v ( a x ) b, f or a x, bay ∈ S ≤ ( ayb )( uxb ayva )( axb ). This implies ( axb )( ayb ) ≤ ( ayb ) t ( axb ) ≤ ( ayb )( ayb ) t ( axb )( axb ) , t = uxb ayva and thus ( axb )( ayb ) ≤ aybsaxb, where s = aybtaxb ∈ aSb . Thus aSb is weakly commuta-tive. (2) ⇒ (3): This is obvious by taking b = a .(3) ⇒ (1): Let a, b, c ∈ S . Then abca, aca ∈ aSa . Since aSa is weakly commutative. Then thereis s ∈ aSa such that ( abca ) aca ≤ acasaabca . Now for aba, abca ∈ aSa , there is t ∈ aSa such that abaabca ≤ abcataba . Thus abca ≤ ( abca )( abca ) ≤ abca ca bca ≤ abca ca ba bca = ( abcaaca )( abaabca ) ≤ ( acasa ca )( abcataba ) ≤ acuba ; where u = asa bca bcata ∈ S . Hence S is normal. Definition 3.19.
An idempotent ordered semigroup ( S, · , ≤ ) is called left normal (right normal) iffor any a, b, c ∈ S, there exists x ∈ S such that abc ≤ acxb ( abc ≤ bxac ) . heorem 3.20. Let S be a left normal idempotent ordered semigroup, then1. L is the least complete semilattice congruence on S ;2. S is a complete semilattice of LZidempotent ordered semigroups.Proof. (1): Let a, b ∈ S such that aρb . Then there are x, y, u, v ∈ S such that a ≤ a ( xbya ) , b ≤ b ( uavb ) . (1)Since S is left normal, we have for x, b, ya ∈ S, xbya ≤ xyatb for some t ∈ S . Similarly there is s ∈ S such that uavb ≤ uvbsa . So from 1, a ≤ ( axyat ) b and b ≤ ( buvbs ) a . Hence a L b . Thus ρ ⊆ L .Again, let a, b ∈ S such that a L b . Thus there re u, v ∈ S such that a ≤ ub and b ≤ va . Bylemma[], we have a ≤ a = aaa ≤ auba ≤ aubba for some u, b ∈ S . Therefore aρb . Thus L ⊆ ρ .Thus L = ρ .(2): Here we are only to proof that each L -class is a left zero. For this let L -class ( x ) L = L, (say) for some x ∈ S . Clearly ( x ) L is a subsemigroup of S . Take a, b ∈ L . Then y, z ∈ S such that a ≤ yb, b ≤ za . Since S is left normal, there is t ∈ S such that a ≤ yb ≤ ( yb ) b ≤ yzab .This implies a ≤ a ≤ a ( ayzb ) b . Thus ( a ) L = ( a yzb ) L = ( ayzb ) L . Therefore L is left zero.Hence S is a complete semilattice of left zero idempotent ordered semigroups. Theorem 3.21.
Let S be a idempotent ordered semigroup, then S is normal if and only if L is rightnormal band congruence and R is left normal band congruence.Proof. First we shall see that L is left congruence on S . For this let us take a, b ∈ S such that a L b and c ∈ S . Then there is x, y ∈ S such that a ≤ xb, b ≤ ya . Now as S is normal idempotent ordered semi-group, ca ≤ cxb ≤ cxbcxb ≤ cxbx ( s ) cb for some s ∈ S . Thus a ≤ s cb where s = cxbxs ∈ S .Again cb ≤ s ca where s = cyays ∈ S . So ca L cb . It finally shows that L is congruence on S .Similarly it can be shown that R is congruence on S .Next consider that a, b, c ∈ S are arbitrary. Then since S is normal idempotent ordered semigroup, abc ≤ abcabc ≤ abcbt ac ≤ acb ( t t bac ) for some acbt t ∈ S. Also bac ≤ bacbac ≤ bacat bc ≤ ( bct t abc ) for some bct t ∈ S . So abc L bac . Similarly abc R acb . This two relations respectivelyshows that L is right normal band congruence and R is left normal band congruence.Conversely, suppose that L is right normal band congruence and R is left normal band congru-ence. Consider a, b, c ∈ S . Then abc R acb and bca L cba . Then ∃ x , x ∈ S such that abc ≤ ( acb ) x and bca ≤ x cba. abc ≤ ( abc ) bca ≤ ( acb ) x bca ≤ ac ( bx x c ) ba f or some bx x c ∈ S . Hence S is a Nidempo-tent ordered semigroup. References [1] A.K.Bhuniya and K. Hansda, Complete semilattice of ordered semigroups, arXiv:1701.01282v1.[2] N. Kehayopulu, Note on Green’s relation in ordered semigroup ,
Math. Japonica , (1991),211-214.[3] N. Kehayopulu, On completely regular poe -semigroups, Math. Japonica (1992), 123-130.[4] N. Kehayopulu, On regular duo ordered semigroups, Math. Japonica , (1992), 535-540.[5] N. Kehayopulu, On intra-regular ordered semigroups, Semigroup Forum , (1993), 271-278.[6] N. Kehayopulu, On completely regular ordered semigroups, Scinetiae Mathematicae (1998),27-32.[7] N. Kehayopulu and M. Tsingelis, Semilattices of Archimedean ordered semigroups,
Algera Col-loquium (2008), 527-540.[8] T. Saito, Ordered idempotent semigroups,
J. Math. Soc. Japan14(2)