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![]() Advances in Pure Mathematics, 2011, 1, 73-76 doi:10.4236/apm.2011.13016 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM A Study on the Conversion of a Sem igroup to a Semilattice Bahman Tabatabaie, Seyed Mostafa Zebarjad Department of Mat hematics, Shiraz University, Shiraz, Iran E-mail: s_zebarjad@yahoo.com Received January 19, 2011; revised March 15, 2011; accepted March 25, 2011 Abstract The main aim of the current research has been concentrated to clarify the condition for converting the inverse semigroups such as S to a semilattice. For this purpose a property the so-called *unitaryE has been de- fined and it has been tried to prove that each inverse semigroups limited with *unitaryE show the specifi- cation of a semilattice. Keywords: Semigroup, Semilattice, *unitaryE 1. Introduction 1.1. Literature Survey Literature survey done by th e authors show that a special class of semigroups possessing is formed by the * E unitary inverse semigroups, sometimes also called 0 *unitaryE, which was defined by Szendrei [1] and has been intensely studied in the semigroup literature. See, for example, Kellendonk’s topological groupoid is Hausdorff when S is *unitaryE[2], and the related class of unitaryE inverse semigroups have also been shown to prov ide Hausdorf f groupoids [3 ]. In the current research the authors try to prove that each inverse semi- groups limited with *unitaryE show the specification of a semilattice. For this purpose, firstly we present ele- mentary concepts as follows. 1.2. Preliminary Definitions and Propositions A groupoid is a set G together with a subs et 2 GGG , a product map ,ab ab. From 2 G to G, and an inverse map 1 aa (so that 1 1 aa ) from G onto G such that: 1) if 2 ,,, ,ab bcG then 2 ,,,ab ca bcG and ab cabc. 2) 12 ,bb G for all bG , and if 2 ,ab G then 1 aabb and 1.ab ba Note that 2 G is nothing but the set of all pairs , x y in GG for which x y is defined, and 2 G is called the set of composable pairs of the groupoid G[3]. If 1 , x Gd xxx is the domain of x and 1 rx xx is its range. The pair , x y is composable if and only if the range of y is the domain of x . 0 GdGrG is the unit space of G, its elements are units in sense that d x xx and rx x [4]. By an inverse semigroup we mean a semigroup S such that for each a in S, there exists a unique ele- ment a in S with the following properties: ,aa aa and aaa a It is well known that the correspondence aa is an involutive anti-homomorphism, i.e., abb a for all a and b in S. It is very common to denote it by ES, the set of all idempotent elements of S, it means that 2 aa for all a in ES. It is clear that aa for all a in ES. A very important example of an inverse semigroup is given by SIX the set of all partial one-to-one maps on a set X . So each element of I X is a bijec- tion form a subset U of X onto another subset V of X . The set I X is a semigroup where the multi- plication rule is given by composition of partial maps with the largest possible domain. For example, if 12 , I X with 11 1 :UV and 22 2 :UV , then 1 122211 21 :VU VU is given by: 121 2.aa ![]() B. TABATABAIE ET AL. Copyright © 2011 SciRes. APM 74 The element 1 is taken to be 1 1 . It is easily checked that I X is an inverse semigroup [3,5]. We recall that a relation on a set X is called a partial ordering of X if for all ,,abc X: 1) aa 2) ab and ba implies ab 3) ab and bc implies ac. The following example is of great importance to us. Define ,efefES to mean .effe e It is clear that is a partial ordering of ES. We shall call the natural partial ordering of ES. An element b of a partially ordered set X is called an upper bound of a subset Y of X , if yb for each y in Y. An upper bound b of Y is called a least upper bound or join of Y, if bc for every up- per bound c of Y. If Y has a join in X , it is clearly unique. Lower bound and greatest lower bound or meet can be defined sim i l a rly. A partially ordered set X is called a semilattice if every two elements subset ,ab of X has a join and a meet in X ; it implies that every finite subset of X has both a join and a meet. The join (or meet) of ,ab will be denoted by ab (or ab)[3]. Definition 1.1 Suppose that S is an inverse semigroup and X can be assumed that as a locally compact Hausdorff topological space. An action of S on X is a semigroup homomorph- ism as follows: :SIX a a such that 1) for every aS there is a continuous a with open dom a i n in X . 2) the union of the domains of all the a coincides with X . Proposition 1.2 Let S be an inverse semigroup, an action of S on a set X and aS, then and aaa a aaaa Proof: Since is an action of S on X then :SIX is a semigroup homomorphism, so for every aS we have aa aa , then aaa a , and simillary a aaa With regard to the above text one may conclude that, 1 a a , and if eES, so e is the iden tity map on its domain. Since the range of each a coincides with the do- main of 1 a a , therefore it can be open as well as its domain. Also it can be mentioned that 1 a , is continu- ous, so a is necessarily a homeomorphism onto its range. For every eES the domain (and range) of e can be denoted by e E, it means: :. ee e EE It is clear to show that the domains of both a and aa is the same, and implies that the domain of a is aa E . Likewise the range of a is given by aa E . Thus : aaa aa EE is a homeomorphism for every aS . Briefly if e and f are in ES then we have ef ef and efef EEE . Proposition 1.3 For each aS and eES we have: ae aa aea EE E Proof: Since N. Sieben [6], R. Ex el [7] and Lawson [8 ] proved it, the author s use their result. Definition 1.4 Let be the subset of SX given by: :aa abSXbE and for every 11 ,ab and 22 ,ab in we will say that 112 2 ,~,abab if 12 bb and there exists an idempotent e in ES such that 1, e bE and 12 ae ae . It is clearly that the relation ~ is an equivalence re- lation on The equivalence class of ,ab will be denoted by ,ab . Let ,: ,GabaSbX and put 2 1122 212 ,,,:a ab abGGGbb And for every 2 112 2 ,,,ab abG define: 1 112 212 2 1 111 1 ,, , ,, a ab abaab aba b it is easy to see that G is a groupoid [3] and the unit space 0 G of G naturally identifies with X under the correspondence 0 ,,eGbbX where e is any idempotent such that S eE. We show G semigroup as ,,GSX . We would now like to give G is a topology. Let aS and U be an open subset of aa E we define ,aU as follows: ,,:aUabGbU The collection of all ,aU is the basis of a topol- ogy on G, and also the multiplication and inversion operations on G are continuous, therefore G is a to- pological grou poid. ![]() B. TABATABAIE ET AL. Copyright © 2011 SciRes. APM 75 2. Main Results Recall from [2] that an inverse semigroup S is natu- rally equipped with a partial order defined by: ab abaaaS Proposition 2.1 Assume that S is an inverse semi- group which is a semilattice. Suppose that is an ac- tion of S on a locally compact Hausdorff space X , such that for each aS, the domain aa Eof a is closed. Then ,,GG SX is Hausdorff. Proof:Suppose ,ac and ,bd are two distinct elements of ,,GSX . The aim is to find two disjoint open subsets 1 T and 2 T of ,,GSX such that: 1212 ,,, ,acTbdT TT We consider two cases: Case 1): If cd: Since X is Hausdorff space then 12121 2 ,open,,,FFXc FdFFF Now let 11 ,aa TaFE and 22 ,bb TbFE Since 1 T and 2 T are open set and 11 ,: , aa TakGkFE 22 ,: , bb TbkGkFE It is clearly that: 1212 ,,, andacTbdTTT Case 2): If cd: Since S is a semilattice let hab so ,, ha hahhac bc hb hbhh Then referring to what proposed in Definition 1.4. hh cE . But hh E is closed then 2\hh TXE can be open and 2 cT. Now we can set T as 2aa bb TE E . But we know that ,:,kTa Tak and it is clear that ,,,,,acaT bcbT . To do so it is enough to prove that ,aT ,bT . Suppose that ,, ,lk aTbT then: ,,,,,~, ,, ,,,,,~, ,, e f lkaTlk aklkak eESk Eaele lkbTlk bklkbk fESkEbflf Since efE S and ef fe, eef kE E , it can be replaced e and f with ef and finally we have: ,aeflefleflfebfebef Therefore we can find an element eES such that ,,. e kEae lele be So leleael le lellellleele , then lea, and similary le b , since hab thus le h , then leleh h , hence lle llehh hh , and finally e lllle hh kEE EE But kT that is contradicts. Definition 2.2 A zero in an inverse semigroup S is an element 0S such that: 00oa aa S Definition 2.3 An inverse semigroup S with zero is said to be unitaryE if for every ,ea S one has that 22 eeaaa . In other words, if an element dominates a nonzero idempotent then that element itself is an idempotent. Proposition 2.4 If S is a unitaryE inverse se- migroup and ,ab belong to the defined semigroup S such that aa bb and ae be for some nonzero idempotent eaa then . ab Proof: We define x aea . So x is nonzero idempotent because: * e aaeaaaae eaaaa Then eaaeaa (because of the ability of idempo- tent elements for being commute) and we have .baxba aeabb beabeaaeax Therefore, we have x ba . Since S is a E unitary which implies that ba is idempotent. Then ba baab so ab is idempotent as well. But, we have bbbb bbbaabab baaa aaaa Setting , y ba b we have that y yb ab ba bb aaaa bb aa bbbb bbb Also y yaa , while ,and ,bbbbby yaaaaay y So it is enough to prove that y ay . We have ayabababbabb babb abby In what follows we give the main result of this paper. Theorem 2.5 In condition that S is a unitaryE inverse semigroup with zero, thencan be appeared as a semilattice. Proof: For proving the above theorem it is necessary to show that ab exists for every ,ab S. If there is not nonzero hS such that ,hab, it is obvious that ![]() B. TABATABAIE ET AL. Copyright © 2011 SciRes. APM 76 0 ab and it can be satisfied for the proof. For doing this we can assume that there is a nonzero hS in which ,hab. Our claim is that .ab bba a Suppose that kaabb and considering to our as- sumption ,,hh aabb we have hh k . Substituting x ak and ybk, 2 2 xx kaakkk x xyy yy kbbkkk also x h hakh hah hhbh hbkh hyh h Using the proposition 2.4 x y will be achieved and so ab baa ab bakxybk baabbbbbaabaa and finally ab bba a (1) By applying the above argument to a,b,h and knowing that 0h and ,hab we have abb baa so ** abb baa and therefore Equation (1) can be modified to 2: bb aaa b (2) We have that ,hab then hahh and hbhh , then we can show that bahh bbhh hh Since S is a unitaryE and ba is dominated by hh , we have 2 baba . By applying the same rea- soning to ,ab and h, 2 ba ba can be a result. Thus * baba ba ba and hence abb babbba ab bbb a (3) By combination of Equations (1) to (3), Equation (4) will be appeared. ab bba abbaaa b (4) At the end we try to prove that ab b can satisfy the following condition ,habbab for every hS such that ,hab. It is clear that ,ab bab and as defined before kaabb , then we have * hh k, and so hahhakh haaab bh hab bhhab bhh Finally habb . It means that abb is the join of a and b and this is the proof of theorem. 3. References [1] M. B. Szendrei, “A Generalization of Mcalister’s P- Theorem for E-Unitary Regular Semigroups,” Acta Sci- entiarum Mathematicarum, Vol. 51, 1987, pp. 229-249. [2] M. V. Lawson, “Inverse Semigroups: The Theory of Par- tial Symmetries,” Word Scientific, Si ngapore, 1998. doi:10.1142/9789812816689 [3] J. Renault, “A Groupoid Approach to C*-Algebra, Lec- ture Notes in Mathematics,” 1st Edition, Sprin- ger-Verlang, Berlin, Vol. 793, 1980. [4] J. M. Howie, “Fundamentals of Semigroup Theory,” Clarendon Press, Oxford, 1995. [5] H. Clifford and G. B. Preston, “The Algebric Theory of Semigroups,” American Mathematical Society, United States, Vol. 1, 1961. [6] N. Sieben, “C*-Crossed Products by Partial Actions and Actions of Inverse Semigroups,” Journal of the Austra- lian Mathematical Society, Series A, Vol. 63, No. 1, 1997, pp. 32-46. [7] R. Exel, “Inverse Semigroups and Combinatorial C*- Algebras,” Bulletin of the Brazilian Mathematical Society, New Series, Vol. 39, No. 2, 2008, pp. 191-313. [8] M. V. Lawson, “Inverse Semigroups, The Theory of Par- tial Symmetries,” Word Scientific, Singapore, 1998. |