Sribala, S.

English

Sribala S.

University of Szeged

On i-ordered inverse semigroups By S. SRI BALA in Madras (India) In this paper we define Z-ordered inverse semigroups (Definition 1) which are partially ordered inverse semigroups whose partial orders are completely determined by families of normal sub-semigroups (Theorem 2). The set of normal sub-semi-groups determining the partial order is analogous to the positive cone in a partially ordered group ([2]). We next consider the set O (A") of all partial o-isomorphism between o-subsets (Definition 6) of a partially ordered set (X, S ) . This set is an inverse sub-semigroup of the symmetric inverse semigroup S(A') on X. A partial order is defined on £)(JQ in a natural way which makes £>(X) a Z-ordered inverse semigroup (Theorem 11). We call O ( Z ) the symmetric Z-ordered inverse semigroup on (X, = ) . Finally, we prove the Preston—Vagner theorem for Z-ordered inverse semigroups claiming that any Z-ordered inverse semigroup can be embedded o-iso-morphically into a symmetric ¿-ordered inverse semigroup of partial o-isomorphisms (Theorem 12). Questions of order theoretic interest will be studied separately and are not dealt with in this paper. Fo r terminology and informat ion on semigroups we refer to [1]. Let S be an inverse semigroup and E the set of idempotents of S: Let I be the lattice of all idempotent separating congruence on 5 with greatest element ¡i. If Mc (e 6 E) is the jU-class containing e, it is known that Me is a normal subgroup of He. D e f i n i t i o n 1. A partially ordered inverse semigroup (5 , is called a I-order-ed inverse semigroup (and s a I - o r d e r on S) if the following conditions hold in S: (1) a^b => a/xb. (2) If Qe={x£S\eSx}, then a'1 QeaQQa-iea for all e£E and a£S. Note that in a X-ordered semigroup no two idempotents are comparable. T h e o r e m 2. Let S be an inverse semigroup and {Qe:e(¿E} a collection of subsets which satisfies the following conditions: (i) Qe is a sub-semigroup of Me containing e and Qe • Qf <= Qef. (ii) a*1 QeaQQa-ieafor all e£E and a£S. (iii) Qc Pi Qe 1 = {e}, where Q¿1 is the set of inverses of elements of Qe in He. Then there exists a partial order ^ on S such that QL,= = and (S, —) a I-ordered inverse semigroup. 208 S. Sribala P r o o f . If a partial order S exists on S such that (S , S ) is a partially ordered inverse semigroup, then it is clear that (S, S ) is a Z-ordered inverse semi-groups. Hence it is enough to establish the existence of such a partial order on S. Let a, b£S. Say a S i apb, ba'^Qaa->> b£Qa-ia. Clearly a^a. Let a,b£S, a^b and b^a. Then a^b => apb, ba~l £Qaa-i, a~ib£Qa-ia; b^a => bpa, ab~ 1 £ €Qbb->' b~x a£Qb-ib. Now, apb => aa~ 1 =bb~1 =e and a~1a = b~ib=f. ba~l and ab~x are inverses of each other in Qc and a~ib and b~1a are inverses of each other in g y and so a = b. Thus S is asymmetric. To show transitivity, let a^b, bSc, ai&b=>apb, ba~l£Qaa-i, a~lbeQa-'a-b^c^bpc, cb~l^Qbb.1, b~lceQb-ib Thus apbpc and aa~l=bb~1 =cc~l =e and a~1a = b~ib = c~i c=f so, ca~l = = cb-^a-1 eQe-Qe i Qe and a~l c = a~l bb'1 c£Qf-Qf a Qf and so a S c . Thus S is a partial order on S. If x£S and e^x, then xpe and x = xe = ex£Qe and so {x£ S\e^x}QQe. On the otherhand if x£QL,, then xpe and xe = ex = x£Qc and so e S x and QeQ g {x6 Thus we have Q e = {xg S ' leSx}. Let a^b, c£ S. Then a^b => apb, ba~l £ Qaa-i, a~l b £ Qa-ia; also apb => acpbc: (bc)(ac)~1 = bcc~la~l = ba~'acc~l a~1 £Qaa-i • Q(acHac)-i g Q(acHac)-i; ( i a c ) ~ ' ( b c ) — : c ~ 1 a ~ 1 b c ^ c ~ 1 Q a - i „ ' C Q 6(oc) - > (m) • Thus ac^bc. Similarly we can show that ca^Scb. Hence (5, S ) is a partially ordered inverse semigroup. It now fol-lows that (S1, S ) is a ^-ordered inverse semigroup. Let (S , S ) be a Z-ordered inverse semigroup. Then, L e m m a 3. a ^ b => b ~ i ^ a ~ l . P r o o f , a^b => anb => a'1 ¡.i\b~{\. Hence b~i =b~* aa~x ^b'1 ba~[ = a~v. P r o p o s i t i o n 4. If T is an inverse sub-semigroup of (S, = ), then (T, S ) is a I-ordered inverse semigroup. P r o o f . Let p0 be the maximum idempotent separating congruence on T and Hr the restriction of p to T. Then pT<gp0. Let Qj = QeDT. Now Qj^ 0<=> e£T. The nonempty sets QJ ( e £ £ f l 7") satisfy the conditions of Theorem 2 and so there exists a partial order S ' on T such that (T, s ' ) is a Z-ordered inverse semigroup. We now show that S coincides with S ' on T. It is immediate that a S i ) (a, bdT) => => a^'b. Conversely if a, b£T then a^'b=>{ap0b, ba-'tQl-,, a~lb£Qj-ta} => => {aa~1 = bb~i = e, a~xa = b~xb=f, ba-^Qe(zMe, a~ib^Qf^MI} ^{apb, a~lb£Qf,: ba-l£Qe}^asb. The following lemma follows immediately. L e m m a 5. (He, S ) , where HL, is the maximal subgroup containing e in (S, S ) , is a partially ordered group. Further, if e2)f, then He and Hf are o-isomorphic. Inverse semigroups 209 We shall now describe the symmetric Z-ordered inverse semigroup on a partially ordered set. Let (X, be a partially ordered set. Let 3 0 0 denote the symmetric inverse semigroup of all partial (1 — 1) transformations on J . If a £ 3 0 0 denote by 4 (a ) and V(a) the domain and the range of a, respectively. D e f i n i t i o n 6. A subset AQ(X, is called an o-subset of X if a£A, aSb (or b^a) implies b£A. It is clear that the set of all o-subsets of (A1, S ) is closed for the operations of set-intersection and set union and contains the null set cp and X. Let © 0 0 denote the subset of 3 0 0 consisting of all o-isomorphisms between o-subsets of X. © (X) contains the map 0 and the identity map of X. P r o p o s i t i o n 7. ©(X) is an inverse sub-semigroup of 3(X). P r o o f . If a £ © 0 0 , then a - 1 is also an o-isomorphism between o-subsets and so belongs to ©GO. Thus it is enough to show that © 0 0 is a subsemigroup of 3 0 0 - Let a, p£S(X) and A=V(a)P\A(P). If A=0, then a/? = 0 £ © 0 0 . If let Ai=Aa~l,A2=Ap. A is an o-subset of X. A1: and A2 are also o-subsets. For, let x £ A j , y ^ x . Since A1 c 4 ( a ) , y £ A ( a ) and so y a S x a £ A . Since A is an o-subset, yadA and so y£Aa~1 = Al. Now At —A(aP), A2=V(aP) and aP:At -+A2 is clearly an o-isomorphism with P~1a~i as its inverse. Thus © 0 0 is an inverse sub-semigroup of 3 0 0 -D e f i n i t i o n s . F o r a , i ?€©(Z) put as=/? <=• a J f P, xaSx j? for all 4(a) and y p - ^ y o c - 1 for a l ly6V(a) . Note that atfpoA(a) = A(p) and.V(a) = V(j0). . P r o p o s i t i o n 9. asp => anP, where n is the maximum idempotent separating congruence on ©00. P r o o f . We will show that a%P implies a~1ea = P~ieP for all idempotents e in © 0 0 which proves that anP ([1] Lemma 7. 57). aS /? => a X P , x a ^ x P for all x£A(a) and yP'^ya'1 for all ^€V(a) . Let A :4(e) Pi 4 (a) = A(e) f)A(P) and B — AsC]A(a) ='A f )4(a) , since e is identity on A. Then e is also identity on B and so ot~1ea is the identity map of Ba and P~lsP is the identity map of Bp. Since a, /?, e are o-isomorphism and a^P we have A and B are.o-subsets and so Ba = Bp. Hence ot~leoi = p-1Ep. P r o p o s i t i o n 10. Let 0 e = { a 6 © O O | e ^ a } , where s is an idempotent of © 0 0 . Then Qt • QCi § Qee,, v~ { Qe v Q Qv-tev for all elements v and idempotents e,sl£ € © 0 0 -P r o o f . Let ot£Qe-, P£QEi. Then s/xa and eiyu/? and so eElfi<xp. Now 4(e) f l 0 4 ( 6 ! ) = 4(eei) = 4(aj3) = V(a/J). x i4 ( e6 ! ) => x S r a and x ^ x P => x ^ x p ^ x a p . y^AieSi) => y a ~ 1 ^ y and y P ~ 1 S y => y P ~ 1 a ~ 1 ^ y . Thus ssli.ioiP, XEE^XCLP and xP~la~i^xee1 for all x£A(e8j) and hence 14 A 210 S. Sribala: Inverse semigroups L e t a c QB and v£0(X). Then v~1evpv~1av. So ,z l (v _ 1 ev) = J ( v ~ 1 a v ) = V(v~1av). x £ ¿ l ( v _ 1 £ v ) and e S a => x v _ 1 £ j ( £ ) f M ( v ) = A(a)f]A(v) => x v _ 1 £ S x v - 1 a in A{v) => x v - 1 £ v S x v _ 1 a v . y£_A(v~leV) => yv~l0L~1S.yv~xz => yv~la.~l v S y v - 1 £ v . Thus v - 1 £ v ^ v _ 1 a v . T h e o r e m 11. (©(A -), is a I-ordered inverse semigroup. It is called the sym-metric I-ordered inverse semigroup on (X, S ) . P r o o f . F rom propositions 9 and 10 it follows that the sets £2c = { a £ CHA^ES a} satisfy the conditions (i) and (ii) of Theorem 2. To show condition (iii) consider a> a~1€Qe- x£á(e), £ S a => X = X E S X O C => x £ a _ 1 S x a a - 1 = x £ ; and E S « " 1 => x = = x s S x a - 1 => x e a S x a - 1 a = x£. Thus we have X E S X C C S X B and x e S x a _ 1 S x £ for all x £ J ( e ) and so a=e=a~l. I t then follows that S is a X-order on €>(X) defined by the sets { 2 J - Hence (D(X), s ) is a Z-ordered inverse semigroup. We now consider representation of Z-ordered inverse semigroups by partial t ransformation. P r o p o s i t i o n 12. If (S, S ) is a I-ordered inverse semigroup, then the mapping oa:x —• xa of Sa~1 onto Sa, belongs to D (S) for every a¿ S. P r o o f . The sets Sa («€ S) are o-subsets of (S, S ) . For , x£ Sa, y^x (or xS.y) => => xpy => Sy= SxQ Sa =>y£Sa. Fur ther Qa and Qa-i are order preserving maps which are inverse of each other. Hence ga£!D(S): T h e o r e m 12. A I-ordered inverse semigroup {S, S ) is o-isomorphic to a I-ordered inverse sub-semigroup of the symmetric I-ordered inverse semigroup ( O ( S ) , s ) . P r o o f . The mapping g: a-+ga of 'S into O ( 5 ) is clearly an isomorphism of the inverse semigroups. We now show that for a, b£S, aSb o QaSgb-a^b=> => apb => Sa~Sb, Sa'^Sb'1 => Qa3^gb. Further , if x £ S a _ 1 , then xga=xa^ ^xb=xgb and if y£Sa, ygii=ygb-i=yb-1^ya-1=>yga-i=yg~1. Hence ga^gb. Next, if Qa^Qb, then S a " 1 = A(el,) = A(c>b)=Sb'1 and Sa = V(ga) = V(gb) = Sb, and for all xc.Sa, xaSxb. Hence aa~1=bb~1=e and a~1a=b~1b=f and so eaS. ^eb-i-eaSb. Hence g is an o-isomorphism of (S, S ) into ( O ( S ) , s ) . References [1] A . H . C L I F F O R D - G . B . PRESTON, Algebraic Theory of Semigroups, Vol. I . & I I , Amer. Math. Soc. Survey 7 ( 1 9 6 1 , 1 9 6 7 ) . [2 ] L . FUCHS, Partially ordered algebraic systems, Pergamon Press ( 1 9 6 3 ) . THE RAMANUJAN INSTITUTE FOR UNIVERSITY OF MADRAS MADRAS-5. INDIA (Received March 30, 1971) 

On Σ-ordered inverse semigroups

http://acta.bibl.u-szeged.hu/14447/1/math_035_207-210.pdf

On Σ-ordered inverse semigroups

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University of Szeged