Groupoids of left quotients

GR OUPOIDS OF LEFT QUOTIENTS N. GHR ODA Abstract. A sub category C of a group oid G is a left order in G , if every elemen t of G can b e written as a − 1 b where a, b ∈ C . A subsemigroup oid C of a group oid G is a left q-order in G , if ev ery element of G can b e written as a − 1 b where a, b ∈ C . W e give a characterization of left orders (q-orders) in group oids. In addition, we describ e the relationship b etw een left I-orders in primitiv e inv erse semigroups and left orders (q-orders) in group oids. 1. Introduction In this article w e inv estigate left orders (q-orders) in group oids. This w ork is part of a con tinuing in v estigation of categories of quotien ts. The motiv ation for our inv estigation comes from semigroups of quotien ts and categories of fractions. Our purp ose is the inv estigation of a similar problem in group oid theory . F oun tain a nd P etrich in troduced the notion of a completely 0-simple semigroup of quotients in [3]. It is w ell-kno wn that group oids are generalisations of groups, also, in v erse semigroups can b e regarded as sp ecial kinds of ordered group oids. The concept of semigroups of quotien ts extends that of a group of quotien ts, in tro duced b y Ore-Dubreil. W e recall that a group G is a gr oup of left quotients of its subsemigroup S if every elemen t of G can b e written as a − 1 b for some a, b ∈ S . Ghro da and Gould [10] extended the classical notion of left orders in inv erse semigroups. They ha v e introduced the following definition: Let Q b e an inv erse semigroup. A subsemigroup S of Q is a left I-or der in Q and Q is a semigr oup of left I-quotient of S , if ev ery elemen t of Q can b e written as a − 1 b where a, b ∈ S and a − 1 is the inv erse of a in the sense of inv erse semigroup theory . The notions of right I-or der and semigr oup of right I-quotients are defined dually . If S is b oth a left and a right I-order in an inv erse semigroup Q , w e say that S is an I-or der in Q and Q is a semigroup of I-quotients of S . If w e insist on a and b b eing R -related in Q , then we say that S is a str aight left I-or der in Q . The theory of categories of fractions was developed b y Gabriel and Zisman [8]. The k ey idea is that starting with a category C we can asso ciate a group oid to Date : Nov ember 26, 2021. Key wor ds and phr ases. primitive in verse semigroup, groupoids of left quotien ts, I-quotients, I-order. 1 GR OUPOIDS OF LEFT QUOTIENTS 2 C by adding all the in v erses of all the elemen ts of C to C . W e then pro duce a group oid G ( C ) = C − 1 C and a functor ι : C − → G such that G ( C ) is generated b y ι ( C ), we call G ( C ) a c ate gory of fr actions . T obias in [20] sho w ed that for any category with conditions which are analogues of the Ore condition in the theory of non-comm utativ e rings (see, [15]), there is a group oid of fractions. No w, we are in a p osition to define a group oid of left quotients. Let C b e a sub category of a groupoid G . W e say that C is a left or der in G or G is a group oid of left quotients of C if ev ery element of G can b e written as a − 1 b for some a, b ∈ C . Right or ders and gr oup oids of right quotients are defined dually . If C is b oth a left and a righ t order in G , then C is an or der in G and G is a gr oup oid of quotients of C . In the case where the category without identities also known as a ‘ quiver ’ or a semigr oup oid , we prop ose the following definition of group oid of q-quotien ts. Let C is a semigroup oid. W e sa y that a group oid G con tains C is a left q- qutients of C or C is a left q-or der in G if every elemen t of G can b e written as a − 1 b for some a, b ∈ C . Right q-or ders and gr oup oids of right q-quotients are defined dually . If C is b oth a left and a righ t q-order in G , then C is a q-or der in G and G is a gr oup oid of q-quotients of C . This work is divided up into eigh t sections. In Section 2 we summarize the bac kground on group oids and inv erse semigroups that w e shall need throughout the article. A Theorem 1.24 in [1] due to Ore and Dubreil sho ws that a semigroup S has a group of left quotien ts if and only if it is right r eversible , that is, S a ∩ S b 6 = ∅ for all a, b ∈ S and S is cancellative. In Section 3 we pro v e the category v ersion of such a theorem. W e stress that this work is not new - it has b een studied by a num b er of authors, by using the notion of category as a collection of ob jects and arrows. W e regard a small category as a generealisation of a moinoid to pro v e such a theorem. Consequen tly , the relationship b etw een the group oids of left quotien ts and in v erse semigroups of left I-quotients b ecomes clearer. In Section 4 w e show that a group oid of left quotients is unique up to iso- morphism. In Section 5 w e establish the connection b etw een group oids of left quotien ts and primitiv e in verse semigroups of left I-quotien ts, then w e sp ecialise our result to left orders in connected group oids in Section 6 a result which ma y b e regarded as a generalisation of Corollary 3.11 in [9], whic h characterised left I- orders in Brandt semigroups. Recall that if we adjoin an elemen t 0 to a group oid G and declare undefined pro ducts in G as b eing equal to zero, we obtain a prim- itiv e in verse semigroup G 0 . W e note that if a category C is a left order in G , then S = C 0 is a left I-order Q = G 0 . In fact, S is a ful l subsemigr oup of Q in the sense that E ( S ) = E ( Q ). In some sense, this justifies introducing the notion of left q-order and left q-quotients. GR OUPOIDS OF LEFT QUOTIENTS 3 The structure of inductiv e group oids that correspond to bisimple inv erse ω - semigroups was determined in [7]. Then it is a natural question to ask for the relationship b etw een left orders in suc h inductive group oids and left I-orders in bisimple in v erse ω -semigroups. In Section 7 we inv estigate such a relationship. Theorem 8.3, in Section 8 giv es necessary and sufficient conditions for a semi- group oid to ha v e a group oid of left q-quotients. 2. Preliminaries and notion In this section w e set up the definitions and results ab out group oids and in verse semigroups. Standard references include [1] for inv erse semigroups, and [12] for group oids. There are tw o definitions of (small) category . The first one in [12] considers the category as a collection of ob jects (sets) and homomorphisms b et w een them satisfying certain conditions. A c ate gory , consisits of a set of ob jects { a, b, c, ... } and homomorphisms b et ween the ob jects such that: ( i ) Homomorphisms are comp osable: giv en homomorphisms a : u − → v and b : v − → w , the homomorphism ab : u − → w exists, otherwise ab is not defined; ( ii ) Comp osition is associative: giv en homomorphisms a : u − → v , b : v − → w , and c : w − → z , ( ab ) c = a ( bc ); ( iii ) Existence of an identit y homomorphism: F or each ob ject u , there is an iden tit y homomorphism e u : u − → u such that for any homomorphism a : u − → v , e u a = a = ae v . A category T is called a sub c ate gory of the category C , if the ob jects of T are also ob jects of C , and the homomorphisms of T are also homorphisms of C suc h that ( i ) for ev ery u in Ob ( T ), the iden tity homomorphism e u is in Hom T ; ( ii ) for every pair of homomorphisms f and g in Hom T the comp osite f g is in Hom T whenever it is defined. The second definition regards the category as an algebraic structure in its o wn righ t. In this definition w e can lo ok at categories as generalisations of monoids. Let C b e a set equipp ed with a partial binary op eration which we shall denote b y · or b y concatenation. If x, y ∈ C and the pro duct x · y is defined we write ∃ x · y . An elemen t e ∈ C is called an identity if ∃ e · x implies e · x = x and ∃ x · e implies x · e = x . The set of iden tities of C is denoted b y C 0 . The pair ( C , · ) is said to be a c ate gory if the following axioms hold: GR OUPOIDS OF LEFT QUOTIENTS 4 (C1): x · ( y · z ) exists if, and only if, ( x · y ) · z exists, in whic h case they are equal. (C2): x · ( y · z ) exists if, and only if, x · y and y · z exist. (C3): F or eac h x ∈ C there exist iden tities e and f such that ∃ x · e and ∃ f · x . It is conv enient to write xy instead of x · y . F rom axiom (C3), it follows that the iden tities e and f are uniquely determined by x . W e write e = r ( x ) and f = d ( x ). W e call d ( x ) the domain of x and r ( x ) the r ange 1 of x . Observe that ∃ xy if, and only if, r ( x ) = d ( y ); in whic h case d ( xy ) = d ( x ) and r ( xy ) = r ( y ). The elemen ts of C are called homomorphisms . A sub c ate gory T of a category C is a collection of some of the identities and some of the homomorphisms of C which include with each homomorphism, a , b oth d ( a ) and r ( a ), and with each comp osable pair of homomorphisms in T , their comp osite. In other w ords, T is a category in its o wn righ t. The tw o definitions are equiv alen t. The first one can b e easily turned in to the second one and vice v ersa. A homomorphism a is said to b e an isomorphism if there exists an element a − 1 suc h that r ( a ) = a − 1 a and d ( a ) = aa − 1 . A gr oup oid G is a category in whic h every elemen t is an isomomorhism. A group may b e though t of as a one-ob ject group oid. A category is c onne cte d if for eac h pair of iden tities e and f there is a homomorphism with domain e and range f . Connected group oids are kno wn as Br andt gr oup oids . If G and P are categories, then ϕ : G − → P is a homomorphis if ∃ xy implies that ( xy ) ϕ = ( xϕ )( y ϕ ) and for all x ∈ G we ha v e that ( d ( x )) ϕ = d ( xϕ ) and ( r ( x )) ϕ = r ( xϕ ). In case where G and P are group oids we hav e that ϕ : G − → P is a homomorphism if ∃ xy implies that ( xy ) ϕ = ( xϕ )( y ϕ ) and so x − 1 ϕ = ( xϕ ) − 1 . The following lemma giv es useful prop erties of group oids whic h will b e used without further mention. Pro ofs can b e found in [14]. Lemma 2.1. L et G b e a gr oup oid. Then for any x, y ∈ G we have ( i ) F or al l x ∈ G we have r ( x − 1 ) = d ( x ) and d ( x − 1 ) = r ( x ) . ( ii ) If ∃ xy , then x − 1 ( xy ) = y and ( xy ) y − 1 = x and ( xy ) − 1 = y − 1 x − 1 . ( iii ) ( x − 1 ) − 1 = x for any x ∈ G . F rom no w on w e shall adopt the second definition of categories. In other w ords, w e regard categories as a generalisation of monoids. 1 Note that in [11] Lawson used the con v erse notion, that is, f = r ( x ) and e = d ( x ) as he comp osed the functions from righ t to left. GR OUPOIDS OF LEFT QUOTIENTS 5 Prop osition 2.2. [18] L et G b e a gr oup and I a non-empty set. Define a p artial pr o duct on I × G × I by ( i, g , j )( j , h, k ) = ( i, g h, k ) and undefine d in al l other c ases. Then I × G × I is a c onne cte d gr oup oid, and every c onne cte d gr oup oid is isomorphic to one c onstructe d in this way. A Br andt semigr oup is a completely 0-simple in verse semigroup. By Theorem I I.3.5 in [18] ev ery Brandt semigroup is isomorphic to B ( G, I ) for some group G and non-empt y set I where B ( G, I ) is constructed as follows: As a set B ( G, I ) = ( I × G × I ) ∪ { 0 } , the binary op eration is defined by ( i, a, j )( k , b, l ) =  ( i, ab, l ) , if j = k ; 0 , else and ( i, a, j )0 = 0( i, a, j ) = 00 = 0 . In [1] it is shown that if we adjoint 0 to a Brandt group oid B , defining xy = 0 if xy is undefined in B , w e get a Brandt semigroup B 0 . Let { S i : i ∈ I } b e a family of disjoint semigroups with zero, and put S ∗ i = S \ { 0 } . Let S = S i ∈ I S ∗ i ∪ 0 with the m ultiplication a ∗ b =  ab, if a, b ∈ S i for some i and ab 6 = 0 in S i ; 0 , else. With this m ultiplication S is a semigroup called a 0-dir e ct union of the S i . An inv erse semigroup S with zero is a primitive inverse semigroup if all its nonzero idempotents are primitiv e, where an idempotent e of S is called primitive if e 6 = 0 and f ≤ e implies f = 0 or e = f . Note that every Brandt semigroup is a primitiv e in v erse semigroup. Theorem 2.3. [18] Br andt semigr oups ar e pr e cisely the c onne cte d gr oup oids with a zer o adjoine d, and every primitive inverse semigr oup with zer o is a 0-dir e ct union of Br andt semigr oups. Notice that a group oid is a disjoint union of its connected comp onen ts. Theorem 2.4. [18] L et G b e a gr oup oid. Supp ose that 0 / ∈ G and put G 0 = G ∪ { 0 } . Define a binary op er ation on G 0 as fol lows: if x, y ∈ G and ∃ x.y in the gr oup oid G , then xy = x.y ; al l other pr o ducts in G 0 ar e 0 . With this op er ation G 0 is a primitive inverse semigr oup. Theorem 2.5. [18] L et S b e an inverse semigr oup with zer o. Then S is primitive if, and only if, it is isomorphic to a gr oup oid with zer o adjoine d. GR OUPOIDS OF LEFT QUOTIENTS 6 In [1], it is sho wn that every primitiv e inv erse semigroup with zero is a 0-direct union of Brandt semigroups. An or der e d gr oup oid ( G , ≤ ) is a group oid G equipp ed with a partial order ≤ satisfies the follo wing axioms: ( OG1 ) If x ≤ y then x − 1 ≤ y − 1 . ( OG2 ) If x ≤ y and x 0 ≤ y 0 and the pro ducts xx 0 and y y 0 are defined then xx 0 ≤ y y 0 . ( OG3 ) If e ∈ G 0 is suc h that e ≤ d ( x ) there exists a unique elemen t ( x | e ) ∈ G , called the r estriction of x to e , such that ( x | e ) ≤ x and d ( x | e ) = e . ( OG3 ) ∗ If e ∈ G 0 is suc h that e ≤ r ( x ) there exists a unique elemen t ( e | x ) ∈ G , called the c or estriction of x to e , suc h that ( e | x ) ≤ x and r ( e | x ) = e . In fact, it is shown in [18] that axiom ( OG3 ) ∗ is a consequence of the other axioms. A partially ordered set X is called a me et semilattic e if, for ev ery x, y ∈ X , there is a greatest low er b ound x ∧ y . An ordered group oid is inductive if the partially ordered set of iden tities forms a meet-semilattice. An ordered group oid G is said to b e ∗ -inductive if each pair of identities that has a low er b ound has a greatest lo w er b ound. W e can lo ok at an y inv erse semigroup as an inductiv e group oid; the order is the natural order and the m ultiplication is the usual multiplication. W e shall now describ e the relationship b et ween inv erse semigroups and induc- tiv e group oids. W e b egin with the follo wing definition. Definition 2.1. F or an arbitrary in verse semigroup S , the r estricte d pr o duct (also called the ‘trace pro duct’) of elemen ts x and y of S is xy if x − 1 x = y y − 1 and undefined otherwise. Let S b e an inv erse semigroup with the natural partial order ≤ . Define a partial op eration ◦ on S as follo ws: x ◦ y defined iff x − 1 x = y y − 1 in which case x ◦ y = xy . Then G ( S ) = ( S, ◦ ) is a group oid and G ( S ) = ( S, ◦ , ≤ ) is an inductiv e groupoid with ( x | e ) = xe and ( e | x ) = ex and e = x − 1 xy y − 1 . Definition 2.2. Let ( G , ., ≤ ) b e an ordered group oid and let x, y ∈ G are such that e = r ( x ) ∧ d ( y ) is defined. Then the pseudopr o duct of x and y is defined as follo ws: x ⊗ y = ( x | e )( e | y ) . GR OUPOIDS OF LEFT QUOTIENTS 7 If ( G , ., ≤ ) is an inductive group oid, then S ( G ) = ( G , ⊗ ) is an inv erse semi- group ha ving the same partial order as G such that the in v erse of any element in ( G , ., ≤ ) coincides with the inv erse of the same elemen t in S ( G ). The pseu- dopro duct is ev erywhere defined in S ( G ) and coincides with the pro duct · in G whenev er · is defined, that is, if ∃ x · y , then x ⊗ y = x · y . It is noted in [18] that in an inductiv e group oid G , for a ∈ G and e ∈ G 0 with e ≤ r ( a ), the corestriction e | a is giv en b y ( e | a ) = ( a − 1 | e ) − 1 . By Linking this with the inv erse semigroup which asso ciated to G , we present a short pro of in the follo wing lemma. Lemma 2.6. L et ( G , ., ≤ ) b e an inductive gr oup oid asso ciate d to an inverse semi- gr oup ( G , ⊗ ) . If a ∈ G and e ∈ G 0 with e ≤ r ( a ) , then ( a − 1 | e ) − 1 = ( e | a ) . Pr o of. First we show that ( a − 1 | e ) exists. As e ≤ r ( a ) and r ( a ) = d ( a − 1 ) w e hav e that e ≤ d ( a − 1 ). Hence by ( OG3 ) ( a − 1 | e ) exists. T o sho w that ( a − 1 | e ) is the in v erse of ( e | a ). W e note that ( a − 1 | e )( e | a )( a − 1 | e ) = ( a − 1 e )( ea )( a − 1 e ) = a − 1 e = ( a − 1 | e ) . Also, ( e | a )( a − 1 | e )( e | a ) = ( ea )( a − 1 e )( ea ) = ea = ( e | a ) .  W e recall that a semigroup Q with zero is defined to b e c ate goric al at 0 if whenev er a, b, c ∈ Q are suc h that ab 6 = 0 and bc 6 = 0, then abc 6 = 0. The set of non-zero elemen ts of a semigroup S will b e denoted b y S ∗ . Let Q b e an inv erse semigroup which is categorical at zero. Define a partial binary op eration ◦ on Q ∗ b y a ◦ b = ( ab, if a − 1 a = bb − 1 ; undefined , otherwise . It is easy to see that (C1) holds. Assume that a ◦ b and b ◦ c are defined in Q ∗ so that ab 6 = 0 and bc 6 = 0 in Q . As Q categorical at 0 we hav e abc 6 = 0 so that a ◦ ( b ◦ c ) is defined in Q ∗ . On the other hand, if a ◦ ( b ◦ c ) exists in Q ∗ , then b − 1 b = cc − 1 and a − 1 a = ( bc )( bc ) − 1 = bcc − 1 b − 1 = bb − 1 . Hence a ◦ b and b ◦ c exist. Th us (C2) holds. F or an y a ∈ Q ∗ the identities d ( a ) = aa − 1 and r ( a ) = a − 1 a satisfy (C3). Hence Q ∗ is a category and an y elemen t a in Q ∗ has the same in v erse a − 1 as in Q . W e ha v e Lemma 2.7. L et Q b e an inverse semigr oup with zer o. If Q c ate goric al at 0 , then Q ∗ = Q \ { 0 } is a gr oup oid. F ollo wing [17], we define Green’s relations on an ordered group oid ( G , · , ≤ ). First w e define useful subsets of G . GR OUPOIDS OF LEFT QUOTIENTS 8 F or a subset H of G , we define ( H ] as follo ws: ( H ] = { t ∈ G : t ≤ h for some h ∈ H } . F or a, b ∈ G , put G a = { xa : x ∈ G and ∃ xa } . W e define a G and a G b similarly . A nonempt y subset I of G is called a right ( left ) ide al of G if (1) I G ⊆ I ( G I ⊆ I ) (2) if a ∈ I and b ≤ a , then b ∈ I . W e say that I is an ide al of G if it is b oth a righ t and a left ideal of G . W e denote by R ( a ) , L ( a ) , I ( a ) the righ t ideal, left ideal, ideal of G , resp ectively , generated b y a ( a ∈ G ). F or eac h a ∈ G , w e ha v e R ( a ) = ( a ∪ a G ] , L ( a ) = ( a ∪ G a ] and I ( a ) = ( a ∪ a G ∪ G a ∪ G a G ] . F or an ordered group oid G , the Gr e en ’s r elations R , L and J defined on G by a R b ⇐ ⇒ R ( a ) = R ( b ); a L b ⇐ ⇒ L ( a ) = L ( b ); a J b ⇐ ⇒ J ( a ) = J ( b ) . It is straigh tforw ard to sho w that a G = d ( a ) G ( G a = G r ( a )) for all a in G . Lemma 2.8. L et G b e an or der e d gr oup oid and let a, b ∈ G . Then (1) a R b ⇐ ⇒ d ( a ) = d ( b ) . (2) a L b ⇐ ⇒ r ( a ) = r ( b ) . Pr o of. Supp ose that R ( a ) = R ( b ) it is clear that if a = b we ha v e that a R b . If a 6 = b , then a ∈ R ( b ) so that a ∈ ( b ∪ b G ] = { t ∈ G : t ≤ h for some h ∈ b ∪ b G } . It is easy to see that aa − 1 ≤ bb − 1 . Hence d ( a ) ≤ d ( b ). Similarly , we can show that d ( b ) ≤ d ( a ). Thus d ( a ) = d ( b ). Con v ersely , supp ose that d ( a ) = d ( b ). Let x ∈ R ( a ) = ( a ∪ a G ] so that x ≤ h for some h ∈ a ∪ a G so that h = a or h ∈ a G = d ( a ) G = d ( b ) G = b G . In the latter case, it is clear that x ∈ R ( b ). In the former case, d ( h ) = d ( a ) and as x ≤ h w e ha ve that xx − 1 ≤ hh − 1 = d ( h ) = d ( a ) = d ( b ) and so x ≤ d ( b ) x . Since d ( b ) x ∈ b G ⊆ b ∪ b G w e hav e that x ∈ R ( b ) and so R ( a ) ⊆ R ( b ). Similarly , R ( b ) ⊆ R ( a ). Thus R ( b ) = R ( a ) as required.  GR OUPOIDS OF LEFT QUOTIENTS 9 3. Left orders in groupoids In this section consider the relationship b et ween left orders in an inductive group oid G and left I-orders in S ( G ). W e give a c haracterisation of left orders in group oids. By using the second definition of categories we pro ve the category v ersion of theorem due to Ore-Dubreil mentioned in the in tro duction. A category is said to b e right ( left ) c anc el lative if ∃ x · a, ∃ y · a ( ∃ a · x, ∃ a · y ) and xa = y a implies x = y ( ax = ay implies x = y ). A c anc el lative c ate gory is one whic h is both left and right cancellative. F ollo wing [11], a category C is said to b e right r eversible 2 if for all a, b ∈ C , with r ( a ) = r ( b ), there exist p, q ∈ C suc h that pa = q b . In diagrammatic terms this just y   x / / b   a / / Let C b e a category and a, b ∈ C such that d ( a ) = d ( b ) w e say that a and b ha v e a pushout , if ax = by for some x, y ∈ C . b   a / / x   y / / Remark 3.1. If a category C is a left order in a group oid G , then any element in G has the form a − 1 b . It is clear that d ( a ) = d ( b ) and a − 1 R a − 1 b L b . If any t w o elements in C ha ve a pushout, then C is a righ t order in G . Hence C is an order in G . W e ha v e the follo wing diagram a / / b   v   q             u / / Lemma 3.1. A c ate gory C is a left or der in an inductive gr oup oid G if and only if ( C , ⊗ ) is a left I-or der in ( G , ⊗ ) . 2 In [11] La wson used d ( a ) = d ( b ) instead of r ( a ) = r ( b ). GR OUPOIDS OF LEFT QUOTIENTS 10 Pr o of. Supp ose that C is a left order in G . F or any q ∈ G , there are a, b ∈ C suc h that for e = aa − 1 bb − 1 w e ha v e q = a − 1 b = a − 1 aa − 1 bb − 1 b = ( a − 1 e )( eb ) = ( ea ) − 1 ( eb ) = ( e | a ) − 1 ( e | b ) = ( a − 1 | e )( e | b ) = a − 1 ⊗ b. It is clear that ( C , ⊗ ) is a subsemigroup of ( G , ⊗ ). The conv erse follo ws b y rev ersing the argumen t.  Lemma 3.2. L et S b e a semigr oup which is a str aight left I-or der in an inverse semigr oup Q . On the set Q define a p artial pr o duct ◦ . Then ( S ∪ E ( Q ) , ◦ ) is a left or der in ( Q, ◦ ) . Pr o of. Supp ose that S is a straight left I-order in Q . F or any q ∈ Q , there are c, d ∈ S such that q = c − 1 d with c R d so that cc − 1 = dd − 1 . Hence q = c − 1 d is defined in ( Q, ◦ ) and c, d ∈ S ∪ E ( Q ). It is easy to see that E ( Q ) = { a − 1 a : a ∈ S } . Let a − 1 a ∈ E ( Q ) for some a ∈ S and let b ∈ S suc h that ba − 1 a is defined in ( Q, ◦ ) so that b − 1 b = a − 1 a . Hence b = bb − 1 b = ba − 1 a ∈ S . Similarly , if a − 1 ab is defined, then a − 1 a = bb − 1 and so b = bb − 1 b = a − 1 ab ∈ S . Th us ( S ∪ E ( Q ) , ◦ ) is a left order in ( Q, ◦ ).  The follo wing corollary is clear. Corollary 3.3. L et S b e a semigr oup which is a str aight left I-or der in an inverse semigr oup Q . On the set Q define a p artial pr o duct ◦ . Then ( S, ◦ ) is a q-left or der in ( Q, ◦ ) . The follo wing lemmas give a c haracterisation for categories whic h are left orders in group oids. The pro ofs of such lemmas are quite straigh tforward and it can b e deduced from [8] and [20], but we giv e it for completeness. Lemma 3.4. L et C b e left or der in a gr oup oid G . Then ( i ) C is c anc el lative; ( ii ) C is right r eversible; ( iii ) any element in G 0 has the form a − 1 a for some a ∈ C . Conse quently, C 0 = G 0 . Pr o of. ( i ) This is clear. ( ii ) Let a, b ∈ C with r ( a ) = r ( b ) so that ab − 1 is defined in G . Since G is a category of left quotients of C , we hav e that ab − 1 = x − 1 y where x, y ∈ C and d ( x ) = d ( y ). Then xa = xab − 1 b = xx − 1 y b = y b. GR OUPOIDS OF LEFT QUOTIENTS 11 ( iii ) Let e b e an iden tit y in G 0 . As C is a left order in G w e ha v e that e = a − 1 b for some a, b ∈ C so that d ( a ) = d ( b ). Since e is identit y and d ( a ) = d ( b ) w e ha v e a = ae = aa − 1 b = d ( b ) b = b. Hence e = a − 1 a = r ( a ) ∈ C 0 so that G 0 ⊆ C 0 . Thus C 0 = G 0 .  Lemma 3.5. Supp ose that G is a gr oup oid of left quotients of C . Then for al l a, b, c, d ∈ C the fol lowing ar e e quivalent: ( i ) a − 1 b = c − 1 d ; ( ii ) ther e exist x, y ∈ C such that xa = y c and xb = y d ; ( iii ) r ( a ) = r ( c ) , r ( b ) = r ( d ) and for al l x, y ∈ S we have xa = y c ⇐ ⇒ xb = y d . Pr o of. ( i ) = ⇒ ( ii ) Supp ose that a − 1 b = c − 1 d for a, b, c, d ∈ C so that r ( a ) = r ( c ) and r ( b ) = r ( d ). By Lemma 3.4, C is right reversible and so there are elements x, y ∈ C such that xa = y c . As, r ( x ) = d ( a ) and r ( y ) = d ( c ) we ha ve ac − 1 = x − 1 xac − 1 = x − 1 y cc − 1 = x − 1 y . Since d ( a ) = d ( b ) and d ( c ) = d ( d ) we ha ve ca − 1 = ca − 1 bb − 1 = cc − 1 db − 1 = db − 1 . Hence db − 1 = ca − 1 = y − 1 x . As d ( x ) = d ( y ) and r ( b ) = r ( d ) w e hav e that xb = y d . ( ii ) = ⇒ ( iii ). It is clear that r ( a ) = r ( c ) and r ( b ) = r ( d ). Let xa = y c and xb = y d . Supp ose that ta = r c for all t, r ∈ C . W e hav e to show that tb = r d . By Lemma 3.4, C is righ t reversible and cancellative. Hence since r ( y ) = r ( r ) and C , it follows that k y = hr for some k , h ∈ C . Now, k xa = k y c = hr c = hta, cancelling in C gives k x = ht . Then htb = k xb = k y d = hr d, again cancelling in C giv es tb = r d . Similarly , tb = r d implies ta = r c , as required. ( iii ) = ⇒ ( i ). Since C is right reversible we hav e that ta = r c for some t, r ∈ C so that tb = r d . Then ac − 1 = t − 1 r = bd − 1 , so that a − 1 b = a − 1 bd − 1 d = a − 1 ac − 1 d = c − 1 d, as required.  La wson has deduced the follo wing theorem from [8]. He has called the groupoid G in suc h a theorem a gr oup oid of fr actions of C . GR OUPOIDS OF LEFT QUOTIENTS 12 Theorem 3.6. [11] L et C b e a right r eversible c anc el lative c ate gory. Then C is a sub c ate gory of a gr oup oid G such that the fol lowing thr e e c onditions hold: ( i ) C 0 = G 0 . ( ii ) Every element of G is of the form a − 1 b wher e a, b ∈ C . ( iii ) a − 1 b = c − 1 d if and only if ther e exist x, y ∈ C such that xa = y c and xb = y d . Pr o of. Our pro of is basically the same as the pro of giv en by T obais [20] in the case of categories as a collections of ob jects and homomorphisms, but our pre- sen tation is sligh tly differen t as w e shall use the second definition of categories. F rom Lemmas 3.4 and 3.5, ( i ) and ( iii ) are clear. T o pro v e ( ii ) supp ose that C is righ t rev ersible and cancellativ e. W e aim to construct a group oid G in whic h C is em b edded as a left order in G . This construction is based on ideas by T obias [20] and Ghro da [9]. Let e G = { ( a, b ) ∈ C × C : d ( a ) = d ( b ) } . Define a relation ( a, b ) ∼ ( c, d ) on e G b y ( a, b ) ∼ ( c, d ) ⇐ ⇒ there exist x, y ∈ C such that xa = y c and xb = y d. W e can represen t this relation by the following diagram a                b   ? ? ? ? ? ? ? ? ? ? ? ? ? x O O y   c _ _ ? ? ? ? ? ? ? ? ? ? ? ? ? d ? ?              Notice that if ( a, b ) ∼ ( c, d ), then r ( a ) = r ( c ) and r ( b ) = r ( d ). Lemma 3.7. The r elation ∼ define d ab ove is an e quivalenc e r elation. Pr o of. It is clear that ∼ is symmetric and reflexive. Let ( a, b ) ∼ ( c, d ) ∼ ( p, q ) , where ( a, b ) , ( c, d ) and ( p, q ) in e G . Hence there exist x, y , ¯ x, ¯ y ∈ C such that xa = y c, xb = y d and ¯ xc = ¯ y p, ¯ xd = ¯ y q . T o show that ∼ is transitiv e, w e hav e to sho w that there are elements z , ¯ z ∈ C suc h that z a = ¯ z p and z b = ¯ z q . GR OUPOIDS OF LEFT QUOTIENTS 13 Since C is right rev ersible and r ( y ) = r ( ¯ x ) there are elements s, t ∈ C suc h that sy = t ¯ x . Hence sxa = sy c = t ¯ xc = t ¯ y p. Similarly , sxb = t ¯ y q as required.  Let [ a, b ] denote the ∼ -equiv alence class of ( a, b ). On G = e G / ∼ we define a pro duct as follows. Let [ a, b ] , [ c, d ] ∈ G . Their pro duct is defined iff r ( b ) = r ( c ). Define [ a, b ][ c, d ] = ( [ xa, y d ] , if xb = y c for some x, y ∈ C ; undefined , otherwise , and so w e ha v e the follo wing diagram y   ? ? ? ? ? ? ? ? ? ? ? x              a              b   ? ? ? ? ? ? ? ? ? ? ? c              d   ? ? ? ? ? ? ? ? ? ? ? Lemma 3.8. The multiplic ation is wel l-define d. Pr o of. Supp ose that [ a 1 , b 1 ] = [ a 2 , b 2 ] and [ c 1 , d 1 ] = [ c 2 , d 2 ] are in G . Then there are elemen ts x 1 , x 2 , y 1 , y 2 in C such that x 1 a 1 = x 2 a 2 , x 1 b 1 = x 2 b 2 , y 1 c 1 = y 2 c 2 , y 1 d 1 = y 2 d 2 . No w, [ a 1 , b 1 ][ c 1 , d 1 ] = [ w a 1 , ¯ w d 1 ] and w b 1 = ¯ w c 1 for some w, ¯ w ∈ C and [ a 2 , b 2 ][ c 2 , d 2 ] = [ z a 2 , ¯ z d 2 ] and z b 2 = ¯ z c 2 for some z , ¯ z ∈ C . It is easy to see that [ a 1 , b 1 ][ c 1 , d 1 ] is defined if and only if [ a 2 , b 2 ][ c 2 , d 2 ] is defined. W e ha v e to pro v e that [ w a 1 , ¯ w d 1 ] = [ z a 2 , ¯ z d 2 ], that is, xw a 1 = y z a 2 and x ¯ w d 1 = y ¯ z d 2 , for some x, y ∈ C . Since w b 1 is defined and d ( a 1 ) = d ( b 1 ) we ha v e that w a 1 is defined and r ( a 1 ) = r ( w a 1 ). Similarly , z a 2 is defined and r ( a 2 ) = r ( z a 2 ). Hence r ( w a 1 ) = r ( w a 2 ), b y the right reversibilit y of C there are elemen ts x, y ∈ C with xw a 1 = y z a 2 . It remains to show that x ¯ w d 1 = y ¯ z d 2 . By Lemma 3.5, xw b 1 = y z b 2 and as GR OUPOIDS OF LEFT QUOTIENTS 14 w b 1 = ¯ w c 1 and z b 2 = ¯ z c 2 w e ha ve that x ¯ w c 1 = y ¯ z c 2 and so x ¯ w d 1 = y ¯ z d 2 , again b y Lemma 3.5.  Lemma 3.9. The multiplic ation is asso ciative. Pr o of. Let [ a, b ] , [ c, d ] , [ p, q ] ∈ G and set X = ([ a, b ][ c, d ])[ p, q ] = [ xa, y d ][ p, q ] where xb = y c for some x, y ∈ C and Y = [ a, b ]([ c, d ][ p, q ]) = [ a, b ][ ¯ xc, ¯ y q ] where ¯ xd = ¯ y p for some ¯ x, ¯ y ∈ C . It is clear that X is defined if and only if Y is defined. W e assume that [ a, b ][ c, d ] and [ c, d ][ p, q ] are defined. Then for some x, y , ¯ x, ¯ y ∈ C we hav e X = [ xa, y d ][ p, q ] = [ sxa, r q ] where sy d = r p for some s, r ∈ C . Y = [ a, b ][ ¯ xc, ¯ y q ] = [ ¯ sa, ¯ r ¯ y q ] where ¯ sb = ¯ r ¯ xc for some ¯ s, ¯ r ∈ C . W e ha v e to sho w that X = [ sxa, r q ] = [ ¯ sa, ¯ r ¯ y q ] = Y . Then b y definition w e need to show that w sxa = ¯ w ¯ sa and w r q = ¯ w ¯ r ¯ y q for some w , ¯ w ∈ C . By cancellativit y in C this equiv alen t to w sx = ¯ w ¯ s and w r = ¯ w ¯ r ¯ y . Since xb and sx are defined we ha ve that sxb is defined and as ¯ sb is defined so that r ( ¯ sb ) = r ( sxb ). By right rev ersibilit y of C w e hav e that w sxb = ¯ w ¯ sb for some w , ¯ w ∈ C . By cancellativity in C we get w sx = ¯ w ¯ s . No w, since w sxb = ¯ w ¯ sb, ¯ sb = ¯ r ¯ xc and xb = y c we hav e that w sy c = ¯ w ¯ r ¯ xc . As C is cancellativ e w e hav e that w sy = ¯ w ¯ r ¯ x so that w sy d = ¯ w ¯ r ¯ xd , but sy d = r p and ¯ xd = ¯ y p so that w r p = ¯ w ¯ r ¯ y p . Thus w r = ¯ w ¯ r ¯ y as required.  F or [ a, b ] ∈ G where xa is defined in C for some x ∈ C , it is clear that [ xa, xb ] ∈ G and d ( x ) xa = xa and d ( x ) xb = xb . Hence we ha ve Lemma 3.10. If [ a, b ] , [ xa, xb ] ∈ G , then [ xa, xb ] = [ a, b ] for al l x ∈ C such that xa is define d in C . Lemma 3.11. The identities of G have the form [ a, a ] wher e a ∈ C . GR OUPOIDS OF LEFT QUOTIENTS 15 Pr o of. Supp ose that e = [ a, b ] is an identit y in G where a, b ∈ C . Let [ m, n ] ∈ G suc h that [ m, n ][ a, b ] is defined and [ m, n ][ a, b ] = [ m, n ] . Then [ xm, y b ] = [ m, n ] for some x, y ∈ C with xn = y a . Hence uxm = v m and uy b = v n for some u, v ∈ C ; cancelling in C giv es that ux = v so that uy b = v n = uxn . Again, b y cancellativit y , it follo ws that xn = y b and as xn = y a we hav e that y b = xn = y a . Using cancellativity in C once more we obtain a = b . Thus e = [ a, a ]. Similarly , if [ a, b ][ m, n ] = [ m, n ] w e ha v e that a = b . It remains to sho w that the iden tit y is unique. Supp ose that [ a, b ][ c, c ] = [ a, b ][ d, d ] = [ a, b ] for some iden tities [ c, c ] , [ d, d ] ∈ G . Then by definition [ xa, y c ] = [ x 0 a, y 0 d ] where xb = y c and x 0 b = y 0 d for some x, y , x 0 , y 0 ∈ C . Hence uxa = v x 0 a and uy c = v y 0 d . By definition of ∼ and Lemma 3.10, [ c, c ] = [ y c, y c ] = [ y 0 d, y 0 d ] = [ d, d ] . As required. Similarly , [ c, c ][ a, b ] = [ d, d ][ a, a ] = [ a, b ] implies that [ c, c ] = [ d, d ].  Supp ose that [ a, b ] ∈ G . Then as d ( a ) a = d ( a ) a w e hav e [ a, a ][ a, b ] = [ d ( a ) a, d ( a ) b ] = [ a, b ] . Similarly , d ( b ) b = d ( b ) b whence [ a, b ][ b, b ] = [ d ( a ) a, d ( a ) b ] = [ a, b ] . Hence d ([ a, b ]) = [ a, a ] and r ([ a, b ]) = [ b, b ]. By the abov e argumen t and Lemma 3.9, the following lemma is clear. Lemma 3.12. G is a c ate gory. If [ a, b ] ∈ G , then it is clear that [ b, a ] ∈ G , as d ( a ) b = d ( a ) b we hav e [ a, b ][ b, a ] = [ d ( a ) a, d ( a ) a ] = [ a, a ] = d ([ a, b ]) Similarly , [ b, a ][ a, b ] = [ b, b ] = r ([ a, b ]). That is, [ b, a ] is the in verse of [ a, b ] in G . Th us w e ha v e Lemma 3.13. G is a gr oup oid. Although from ( i ) w e know that C 0 = G 0 , but in the following lemma we give a new proof dep ends on the structure of G . Lemma 3.14. C 0 = G 0 . GR OUPOIDS OF LEFT QUOTIENTS 16 Pr o of. Supp ose that e = [ a, b ] is an iden tit y in G where a, b ∈ C . Then [ a, b ] = [ d ( x ) , x ] − 1 [ d ( y ) , y ] . As [ d ( x ) , x ][ d ( x ) , x ] − 1 is defined, so that [ d ( x ) , x ][ a, b ] is defined and since [ a, b ] is an iden tit y w e ha v e that [ d ( x ) , x ][ a, b ] = [ d ( x ) , x ]. Hence [ d ( x ) , x ] = [ d ( x ) , x ][ a, b ] = [ d ( x ) , x ][ d ( x ) , x ] − 1 [ d ( y ) , y ] = [ d ( y ) , y ] . Th us x = y and so a = b . But [ a, a ] = [ r ( a ) , r ( a )] = [ d ( r ( a )) , r ( a )] so that the iden tities in C hav e the form [ d ( r ( a )) , r ( a )] for an y a ∈ C .  Lemma 3.15. The mapping θ : C − → G define d by aθ = [ d ( a ) , a ] is an emb e d- ding of C in G . Pr o of. It is clear that θ is w ell-defined. T o show that θ is one-to-one, let [ d ( a ) , a ] = [ d ( b ) , b ] so that ua = v b and u d ( a ) = v d ( b ) for some u, v ∈ C . Hence a = b . Let a, b ∈ C such that ab is defined. W e ha ve aθ bθ = [ d ( a ) , a ][ d ( b ) , b ] = [ u d ( a ) , v b ] where ua = v d ( b ) for some u, v ∈ C = [ u d ( a ) , uab ] as ua = v d ( b ) = v = [ d ( a ) , ab ] b y lemma 3.10 = [ d ( ab ) , ab ] as d ( a ) = d ( ab ) = ( ab ) θ . Th us θ is a homomorphism.  By Lemma 3.15, we can consider C as a sub category of G . Let [ a, b ] ∈ G and aθ = [ d ( a ) , a ], bθ = [ d ( b ) , b ]. As d ( a ) = d ( b ) w e ha ve that aθ = [ d ( a ) , a ] and bθ = [ d ( a ) , b ]. Hence ( aθ ) − 1 ( bθ ) = [ d ( a ) , a ] − 1 [ d ( a ) , b ] = [ a, d ( a )][ d ( a ) , b ] = [ ua, v b ] where u = u d ( a ) = v d ( a ) = v = [ ua, ub ] b y Lemma 3.10 = [ a, b ] . Hence C is a left order in G . This completes the pro of of Theorem 3.6.  Corollary 3.16. A sub c ate gory C is a left or der in a gr oup oid G if and only if C is right r eversible and c anc el lative. Pr o of. If C is a left order a groupoid G , then b y Lemmas 3.4, C is righ t rev ersible and cancellativ e. Con v ersely , if C is right rev ersible and cancellativ e, then by ( ii ) in Theorem 3.6, C is a left order a group oid.  GR OUPOIDS OF LEFT QUOTIENTS 17 4. Uniqueness In this section we show that a category C has, up to isomomorphism, at most one group oid of left I-quotients. Theorem 4.1. L et C b e a left or der in gr oup oid G . If ϕ is an emb e dding of C to a gr oup oid T , then ther e is a unique emb e dding ψ : G − → T such that ψ | S = ϕ . Pr o of. Define ψ : G − → T b y ( a − 1 b ) ψ = ( aϕ ) − 1 ( bϕ ) for a, b, c, d ∈ C . Supp ose that a − 1 b = c − 1 d so that xa = y c and xb = y d for some x, y ∈ C , b y Lemma 3.5. Hence xϕaϕ = y ϕcϕ and xϕbϕ = y ϕdϕ in C ϕ . Th us aϕcϕ − 1 = xϕ − 1 y ϕ = bϕdϕ − 1 so that aϕ − 1 bϕ = cϕ − 1 dϕ. It follo ws that ψ is well-defined and 1-1. It remains for us to sho w that ψ is a homomorphism. Let a − 1 b, c − 1 d ∈ G where a, b, c, d ∈ C . No w, ( a − 1 bc − 1 d ) ψ = (( xa ) − 1 ( y d )) ψ = ( xa ) ϕ − 1 ( y d ) ϕ = aϕ − 1 xϕ − 1 y ϕdϕ, where xb = y c for some x, y ∈ C . W e ha v e that xϕbϕ = y ϕcϕ and so bϕcϕ − 1 = xϕ − 1 y ϕ . Hence ( a − 1 bc − 1 d ) ψ = aϕ − 1 xϕ − 1 y ϕdϕ = aϕ − 1 bϕcϕ − 1 dϕ = ( a − 1 b ) ψ ( c − 1 d ) ψ . Finally , to see that ψ is unique, supp ose that θ : G − → T is an embedding with θ | S = ϕ . Then for an elemen t a − 1 b of G , we hav e ( a − 1 b ) θ = ( a − 1 θ )( bθ ) = ( aθ ) − 1 ( bθ ) = ( aϕ ) − 1 ( ϕ ) = ( a − 1 b ) ψ so that θ = ψ .  The follo wing corollary is straightforw ard. Corollary 4.2. If a c ate gory C is a left or der in gr oup oids G and P , then G and P ar e isomorphic by an isomorphism which r estricts to the identity map on C . GR OUPOIDS OF LEFT QUOTIENTS 18 5. Primitive inverse semigroups of left I-quotients and groupoids of left quotients In this section w e are concerned with the relationship betw een primitiv e in v erse semigroups of left I-quotients and group oids of left quotien ts. Let Q b e a primitiv e inv erse semigroup. By using the restriction pro duct of elemen ts of Q w e can asso ciate a group oid G to Q . On the other hand, if G is a group oid, then G 0 is a primitive inv erse semigroup. In fact, an y primitiv e in v erse semigroup is isomorphic to one constructed in this w a y . In particular, primitiv e in v erse semigroups of left I-quotien ts and group oids of left quotients are equiv alent structures in the sense that each can b e reconstructed from the other. It is w ell-kno wn that any primitiv e in verse semigroup is categorical at 0. W e sa y that, S is 0-c anc el lative if b = c follows from ab = ac 6 = 0 and from ba = ca 6 = 0. In [9] w e studied left I-orders in primitive in verse semigroups b y using the relation λ on any semigroup with zero whic h defined as follo ws: a λ b if and only if a = b = 0 or S a ∩ S b 6 = 0 . The follo wing theorem giv es necessary and sufficien t conditions for a semigroup to ha v e a primitiv e in v erse semigroup of left I-quotients. Theorem 5.1. [9] A semigr oup S is a left I-or der in a primitive inverse semi- gr oup Q if and only if S satisfies the fol lowing c onditions: ( A ) S is c ate goric al at 0 ; ( B ) S is 0-c anc el lative; ( C ) λ is tr ansitive; ( D ) S a 6 = 0 for al l a ∈ S ∗ . Supp ose that a semigroup S has a primitiv e in v erse semigroup of left I-quotien ts Q . By Prop osition 2.4 of [9], S con tains a zero. Define a partial binary op eration ◦ on G = Q ∗ b y a ◦ b = ( ab, if ab 6 = 0; undefined , otherwise . Notice that in Q , for any non-zero elements a and b in Q w e hav e that ab 6 = 0 if and only if a − 1 a = bb − 1 . W e ha ve a group oid G with G 0 = E ( Q ) ordered by the natural partial order on Q . Hence G is inductive. If a semigroup S is a left I-order in Q , then S satisfies the conditions of Theo- rem 5.1. Since Q categorical at 0, it follows that Q ∗ with the partial pro duct is a group oid, by Lemma 2.7. It is clear that an y element in Q ∗ can b e written as a − 1 b where a, b ∈ S ∗ . Unfortunately , S ∗ is not a sub category of Q ∗ as w e do not insist on S b eing full in Q . By adding E ( Q ) to S ∗ w e ha v e that C = S ∗ ∪ E ( Q ) GR OUPOIDS OF LEFT QUOTIENTS 19 is a sub category of Q ∗ . F or, let a ∈ S ∗ and f ∈ E ( Q ), if af is defined in C , then af 6 = 0 in Q . By Lemma 2.1 in [4] w e hav e a − 1 a = f and so a = aa − 1 a = af so that a = af ∈ C . Similarly , if ea is defined in C , then a = ea . Th us C is a left order in Q ∗ . W e ha ve Lemma 5.2. If a semigr oup S is a left I-or der in a primitive inverse semigr oup Q , then C = S ∗ ∪ E ( Q ) is a left or der in the gr oup oid Q ∗ . Corollary 5.3. If a semigr oup S is a left I-or der in a primitive inverse semi- gr oup Q , then C = S ∗ ∪ E ( Q ) is a right r eversible c anc el lative sub c ate gory of the gr oup oid G = Q ∗ asso ciate d to Q . No w, in case we did not drop the zero from Q . Under the same multiplication w e can show that C = ( S ∪ E ( Q ) , ◦ ) is a left order in G = ( Q, ◦ ) where G is inductiv e and { 0 } is an isolated identit y of G , in the sense that it is not a domain or co domain of an y hommorphism. Also, G ∗ = G \ { 0 } is an *-inductiv e group oid with G ∗ 0 = E ( Q ) ∗ . W e hav e shown ho w to construct a group oid of left quotien ts from a primitiv e in v erse semigroup of left I-quotients. The rest of this section is dev oted to sho wing ho w to construct a primitiv e in v erse semigroup of left I-quotien ts from a groupoid of left quotien ts. Supp ose that C is a left order in a group oid G . Define m ultiplication on Q as: ab = ( a ◦ b, if ∃ a ◦ b in G ; 0 , otherwise . By Theorem 3.16, C is right reversible and cancellative. Put S = C ∪ { 0 } and Q = G ∪ { 0 } . By Theorem 2.4, Q is a primitiv e inv erse semigroup. It is clear that S is a subsemigroup of Q . Moreov er, it is a left I-order in Q . Thus w e hav e Lemma 5.4. If a c ate gory C is a left or der in a gr oup oid G , then S = C ∪ { 0 } is a left I-or der in Q = G ∪ { 0 } . Notice that S in the ab o v e lemma is full, as C and G ha v e the same set of the iden tites. 6. Connected groupoids of left quotients In this section w e giv e necessary and sufficien t conditions for a category to ha ve a connected group oid of left quotients. That is, we sp ecialise our result in the previous section to left orders in connected group oids. In general a group oid it migh t contains a zero, but a non-trivial connected group oid can not ha v e a zero. Let C b e a category . W e sa y that C satisfies the c onne cte d c ondition if, for an y a, b ∈ C there exist c, d ∈ C with d ( c ) = d ( d ) such that r ( c ) = r ( a ) and GR OUPOIDS OF LEFT QUOTIENTS 20 r ( d ) = r ( b ). W e regard the connected condition in diagrammatic terms as follo ws. a / / d   ? ? ? ? ? ? ? ? ? ? ? c ? ?            b / / Let C b e a left order in a group oid G . It is clear that if C is connected, then G is connected. Also, if C is connected, then it has the connected condition, but the con v erse is not true. Lemma 6.1. A right r eversible, c ate gory C is a left or der in a c onne cte d gr oup oid G if and only if C satisfies the c onne cte d c ondition. Pr o of. Supp ose that C is a left order in a connected group oid G . T o sho w that C satisfies the connected condition. Supp ose that a − 1 a and b − 1 b are t wo iden tit y elemen ts of G for some a, b ∈ C . Hence there is an isomomorphism h in G such that d ( h ) = a − 1 a and r ( h ) = b − 1 b . As C is a left order in G , w e ha ve that h = s − 1 t for some s, t ∈ C . Hence r ( s ) = d ( h ) = r ( a ) and r ( t ) = r ( h ) = r ( b ). Th us the connected condition holds on C . Con v ersely , supp ose that C satisfies the connected condition. By Corollary 3.16, C is a left order in a group oid G . W e pro ceed to show that G is connected. Sup- p ose that e and f are tw o iden tit y elemen ts of G so that e = a − 1 a and f = b − 1 b for some a, b ∈ C , b y Lemma 3.4. By assumption there exist c, d ∈ C with d ( c ) = d ( d ) suc h that r ( b ) = r ( d ) and r ( c ) = r ( a ). As d ( c ) = d ( d ) w e ha v e c − 1 d is defined in G and d ( c − 1 d ) = c − 1 c = a − 1 a and r ( c − 1 d ) = d − 1 d = b − 1 b . Thus G is connected.  Lemma 6.2. L et G = S i ∈ I G i b e a gr oup oid wher e G i is a c onne cte d gr oup oid. If C is a left or der in G , then C is a disjoint union of c ate gories that ar e left or ders in the c onne cte d gr oup oids G i ’s. Pr o of. Supp ose that C is a left order in G . Then ev ery elemen t of G can b e written as a − 1 b where a, b ∈ C . As G is a disjoin t union of G i ’s we hav e that a − 1 , b ∈ G i for some i ∈ I . Since G i is a group oid w e ha ve that a, b ∈ G i . Hence C i = C ∩ G i 6 = ∅ . It is clear that C i is a sub category of G i . W e claim that C i is a left order in G i . Let q = a − 1 b ∈ G i so that a, b ∈ C ∩ G i = C i . It is clear that C is a disjoint union of the categories C i , i ∈ I .  GR OUPOIDS OF LEFT QUOTIENTS 21 It is w ell-kno wn that Brandt semigroups are precisely the connected group oids with a zero adjoined. On the other hand, from any Brandt semigroup w e can reco v er a connected group oid. W e will be using the same tec hnique that we used in the previous section to determine the relationship b etw een left I-orders in Brandt semigroups and left orders in connected group oids. Brandt semigroups are primitive so that the most part of the task has b een done in the previous section. Theorem 6.3. [9] A semigr oup S is a left I-or der in a Br andt semigr oup Q if and only if S satisfies the fol lowing c onditions: ( A ) S is c ate goric al at 0 ; ( B ) S is 0-c anc el lative; ( C ) λ is tr ansitive; ( D ) S a 6 = 0 for al l a ∈ S ∗ ; ( E ) for al l a, b ∈ S ∗ ther e exist c, d ∈ S such that ca R ∗ d λ b . In order to generalise the ab ov e theorem to the category ver sion, we need to obtain a corresp onding condition to (E) to make G connected, that is, we need to sho w that there is an isomorphism b etw een any tw o identities. Assume that a semigroup S has a Brandt semigroup of left I-quotien ts Q . Put C = S ∗ ∪ E ( Q ) and G = Q ∗ . Define multiplication on G as defined b efore Lemma 5.2. By Lemma 5.2, C is a left order in G which is an inductiv e groupoid. W e claim that G is connected. Supp ose that a − 1 a and b − 1 b are tw o identit y elemen ts of G for some a, b ∈ C . Since S is a left I-order in Q , it follows that for a, b ∈ S there exist c, d ∈ S suc h that ca R ∗ d λ b , b y (E). As ca 6 = 0 in S we hav e that ca is defined in C and so c − 1 c = aa − 1 . Since ca R d in Q , by Lemma 2.4 in [9]. W e ha ve d ( ca ) = d ( c ) = cc − 1 = c ( c − 1 c ) c − 1 = c ( aa − 1 ) c − 1 = ( ca )( ca ) − 1 = dd − 1 = d ( d ) . As d λ b in S so that xd = y b 6 = 0 for some x, y ∈ S . Hence xd and y b are defined in C and xd = y b in C so that r ( b ) = r ( d ). It is clear that r ( ca ) = r ( a ). By Lemma 6.1, G is connected. W e ha v e established our claim. W e ha v e Lemma 6.4. If a semigr oup S is a left I-or der in a Br andt semigr oup Q , then C is a left or der in G wher e C = S ∗ ∪ E ( Q ) and G = Q ∗ . No w, w e aim to turn left orders in connected groupoids to Brandt semigroups of left I-quotien ts. Supp ose that C is a left order in a connected group oid G . Let S = C ∪ { 0 } and Q = G ∪ { 0 } . Define multiplication on G as it defined b efore Lemma 5.4. By Lemma 5.4, S is a left I-order in Q where Q is a primitiv e inv erse semigroup. W e GR OUPOIDS OF LEFT QUOTIENTS 22 claim that S satisfies (E). In other w ords, we claim that Q is a Brandt semigroup. Let a, b ∈ S so that a − 1 a, b − 1 b ∈ E ( Q ) = G 0 as G is connected, there are c, d ∈ C suc h that d ( c ) = d ( d ) , r ( a ) = r ( c ) and r ( b ) = r ( d ). As d ( ca ) = d ( c ) = d ( d ) in G we hav e that ca R d in Q so that ca R ∗ d in S , b y Lemma 2.4 in [9]. Since C is a left order in G we ha v e that C is right reversible and as r ( b ) = r ( d ) we hav e that xb = y d is defined for some x, y ∈ C so that xb = y d 6 = 0 in S and so d λ b . Th us w e ha v e established our claim. Hence w e ha ve Lemma 6.5. If a c ate gory C is a left or der in a c onne cte d gr oup oid G , then S = C ∪ { 0 } is a left I-or der in a Br andt semigr oup Q = G ∪ { 0 } . 7. Inductive ω -groupoids of left quotients In this section we are concerned with a sp ecial class of connected group oids of left quotients. This class is asso ciated to a bisimple inv erse ω -semigroup. W e b egin b y describing the w ell kno wn Bruck-R eil ly extension . Let G b e a group and θ b e an endomorphism of G . The Bruck-Reilly exten- sion B R ( G, θ ) of G with resp ect to θ is the set N 0 × G × N 0 with the binary op eration: ( m, a, n )( p, b, q ) =  m − n + s, ( aθ s − n )( bθ s − p ) , q − p + s  where s = max( n, p ). The idemp otents of B R ( G, θ ) are the elements of the form ( n, 1 G , n ) where n ∈ N 0 . Theorem 7.1. [1] Every bisimple inverse ω -semigr oup is isomorphic to some Bruck-R eil ly extension of a gr oup G determine d by an endomorphism of G . W e denote ( ω , ≤ ) the p oset consisting of the natural num b ers under the dual of the usual partial order. F rom Prop osition 2.2, w e kno w that any connected groupoid has the form I × G × I where I is a nonempt y set and G is a group. If we chose I to b e N 0 , then we obtain a connected group oid T = N 0 × G × N 0 . The identities of T hav e the form ( i, 1 , i ) and we hav e the ω -ordering on the identities as follo ws ( a, 1 , a ) ≤ ( b, 1 , b ) ⇐ ⇒ a ≥ b, that is, (0 , 1 , 0) ≥ (1 , 1 , 1) > (2 , 1 , 2) > (3 , 1 , 3) > .... It is sho wn in [7], that T is inductive, and it is called an inductive ω -gr oup oid . F or ( a, g , b ) ∈ T w e hav e that d (( a, g , b )) = ( a, 1 , a ) and r (( a, g , b )) = ( b, 1 , b ). It is sho wn in [7] that with a bisimple inv erse ω -semigroup Q w e can asso ciate an inductiv e ω -group oid isomomorhic to one asso ciated to a Bruck-Reilly extension. GR OUPOIDS OF LEFT QUOTIENTS 23 A natural question to ask at this p oin t is, what are the necessary and sufficient conditions of a category to hav e an inductive ω -group oid of left quotients. By Lemmas 6.1 and Lemma 3.11, the first part of the follo wing lemma is clear. Lemma 7.2. A c ate gory C is a left or der in an inductive ω -gr oup oid T if and only if C satisfies the fol lowing c onditions: ( i ) C is right r eversible; ( ii ) C is c anc el lative; ( iii ) C has the c onne cte d c ondition; ( iv ) C 0 is an ω -chain. Pr o of. Supp ose that C satisfies the Conditions ( i ) − ( iv ). By Corollary 3.16 and Lemma 6.1, we hav e that C is a left order in a connected group oid G . By Lemma 3.4, G has the same iden tities as C . Hence G is an inductive ω -group oid, b y ( iv ). Con v ersely , If C is a left order in an inductive ω -group oid T , then by Corol- lary 3.16, C is right rev ersible and cancellativ e. By Theorem 3.6, C 0 is an ω -chain. Since T is connected w e ha v e that C has the connected condition, by Lemma 6.1.  F ollo wing [9], let B b e the bicyclic monoid. Consider a semigroup S together with a homomorphism ϕ : S − → B . W e define functions l , r : S − → N 0 b y aϕ =  r ( a ) , l ( a )  . W e also put H i,j = ( i, j ) ϕ − 1 , so that S is a disjoint union of subsets of the H i,j and H i,j = { a ∈ S : r ( a ) = i, l ( a ) = j } . It is w ell known that H is a congruence on any bisimple in v erse ω -semigroup Q and Q/ H ∼ = B where B is the bicyclic semigroup. Let ϕ : Q − → B b e a surjective homomorphism with K er ϕ = H . As ab o v e we will index the H -classes of Q by putting H i,j = ( i, j ) ϕ − 1 . Let S b e a left I-order in Q . Let ϕ = ϕ | S so that ϕ is a homomorphism from S to B . Unfortunately , this homomorphism is not surjectiv e in general, since S need not intersect every H -class of Q . But w e can as ab ov e index the elements of S . Theorem 7.3. [9] A semigr oup S is a left I-or der in a bisimple inverse ω - semigr oup Q if and only if S satisfies the fol lowing c onditions: ( A ) Ther e is a homomorphism ϕ : S − → B such that S ϕ is a left I-or der in B ; ( B ) F or x, y , a ∈ S , ( i ) l ( x ) , l ( y ) > r ( a ) and xa = y a implies x = y , ( ii ) r ( x ) , r ( y ) > l ( a ) and ax = ay implies x = y . GR OUPOIDS OF LEFT QUOTIENTS 24 ( C ) F or any b, c ∈ S , ther e exist x, y ∈ S such that xb = y c wher e x ∈ H r ( x ) ,r ( b ) − l ( b )+ max  l ( b ) ,l ( c )  , y ∈ H r ( x ) ,r ( c ) − l ( c )+ max  l ( b ) ,l ( c )  . Supp ose that S is a left I-order in a bisimple inv erse ω -semigroup Q . Let T b e an inductiv e ω -group oid asso ciated to Q . The restricted pro duct ◦ is defined on Q b y the rule that p ◦ q = ( pq , if p − 1 p = q q − 1 ; undefined, otherwise, . Then T = ( Q, ◦ ) is an inductive group oid. It is clear that C = ( S ∪ E ( Q ) , ◦ ) is a sub category of T . F or if, a ∈ S and e ∈ E ( Q ) and e ◦ a ( a ◦ e ) is defined in T , then e = aa − 1 ( a − 1 a = e ) so that e ◦ a = a ( a ◦ e = a ) ∈ C . It is clear that C is a left order in T . W e ha v e Lemma 7.4. If a semigr oup S is a left I-or der in a bisimple inverse ω -semigr oup Q , then C = ( S ∪ E ( Q ) , ◦ ) is a left or der in the inductive gr oup oid T = ( Q, ◦ ) . If a category C is a left order in an inductiv e ω -group oid T , then by Lemma 3.2, ( C , ⊗ ) is a left I-order in ( T , ⊗ ). Hence ( T , ⊗ ) is a bisimple inv erse ω -semigroup so that ( C , ⊗ ) satisfies conditions (A), (B) and (C). Lemma 7.5. If a c ate gory C is a left or der in an inductive ω -gr oup oid T , then the semigr oup ( C , ⊗ ) is a left I-or der in ( T , ⊗ ) . 8. Left q-orders in groupoids In Section 5 w e sho w ed that if a category C is a left order in a group oid G , then S = C 0 is a left I-order in the primitive inv erse semigroup Q = G 0 . The problem is, S is a full subsemigroup of Q as C and G hav e the same set of iden tities. T o solv e this problem we need to delete the iden tites of C . In other w ords, w e need to consider semigroupids. This section is entirely dev oted to proving Theorem 8.3 which gives a c harac- terisation of semigroupoids which hav e a group oid of left q-quotien ts. W e recall that a semigroup oid C is given b y ( a ) a set Ob ( C ) of ob jects; ( b ) for eac h pair ( a, b ) of ob jects, a set C ( a, b ) of homomorphisms; ( c ) for eac h triple ( a, b, c ) of ob jects, a mapping from C ( a, b ) × C ( b, c ) in to C ( a, c ) whic h asso ciates to each u ∈ C ( a, b ) and v ∈ C ( b, c ) the asso ciativ e comp osition uv ∈ C ( a, c ). W e sa y that u ∈ C ( a, b ) has domain dom ( u ) = a and c o domain co d ( u ) = b . W e write Hom ( C ) = S a,b ∈ Ob ( C ) C ( a, b ). A semigroup oid is c onne cte d if C ( u, v ) 6 = ∅ for all u, v ∈ C . W e recall that a semigroup oid is said to b e right c anc el lative if ∃ x · a, ∃ y · a and xa = y a implies GR OUPOIDS OF LEFT QUOTIENTS 25 x = y . A left c anc el lative semigr oup oid is defined dually . A c anc el lative semi- gr oup oid is one which is b oth left and right cancellativ e. It is noted in [16] that for a semigroup oid C one can adjoint a new element 0 not in C such that C ∪ { 0 } is a semigroup with multiplication ab =  the C -pro duct ab if a, b ∈ C and co d ( a ) = dom ( b ); 0 , otherwise. Moreo v er, C ∪ { 0 } is categorical at 0. A semigroup oid C is said to b e right r eversible if for all a, b ∈ C , with co d ( a ) = co d ( b ), there exist p, q ∈ C such that pa = q b where co d ( a ) and co d ( b ) are co domain a and b resp ectiv ely . F rom Lemmas 3.4 and 3.5, w e can easily deduce the following lemmas. The pro ofs are clear and will b e omitted. Lemma 8.1. L et C b e left or der in a gr oup oid G . Then ( i ) C is c anc el lative; ( ii ) C is a right r eversible; Lemma 8.2. Supp ose that G is a gr oup oid of left quotients of C . Then for al l a, b, c, d ∈ C the fol lowing ar e e quivalent. ( i ) a − 1 b = c − 1 d ; ( ii ) ther e exist x, y ∈ C such that xa = y c and xb = y d ; ( iii ) c o d ( a ) = c o d ( c ) , c o d ( b ) = c o d ( d ) and for al l x, y ∈ C we have xa = y c ⇐ ⇒ xb = y d . W e no w state the main result of this section. Theorem 8.3. A subsemigr oup oid C is a left or der in a gr oup oid G if and only if satisfies the fol lowing c onditions: ( A ) C is right r eversible; ( B ) C c anc el lative; ( C ) for al l a ∈ C ther e exists x ∈ C such that xa is define d. Pr o of. If C is a left order in G , then b y Lemmas 8.1, C is right reversible and cancellativ e. Hence ( A ) and ( B ) hold. It remains to prov e ( C ), let a b e any elemen t in C . As C is a left q-order in G w e ha ve that a = x − 1 y for some x, y ∈ G . Thus xa = xx − 1 y = y is defined in C as required. Con v ersely , we supp ose that C satisfies Conditions (A)-(C). W e aim to con- struct a groupoid G in which C is em b edded as a left q-order in G . Let e G = { ( a, b ) ∈ C × C : dom ( a ) = dom ( b ) } . GR OUPOIDS OF LEFT QUOTIENTS 26 Define a relation ( a, b ) ∼ ( c, d ) on e G b y ( a, b ) ∼ ( c, d ) ⇐ ⇒ there exist x, y ∈ C such that xa = y c and xb = y d. Notice that if ( a, b ) ∼ ( c, d ), then co d ( a ) = co d ( c ) and co d ( b ) = co d ( d ). The follo wing lemma is clear. Lemma 8.4. The r elation ∼ define d ab ove is an e quivalenc e r elation. Let [ a, b ] denote the ∼ -equiv alence class of ( a, b ). On G = e G / ∼ we define a pro duct b y [ a, b ][ c, d ] = ( [ xa, y d ] if co d ( b ) = co d ( c ) and xb = y c for some x, y ∈ C ; undefined otherwise , and so w e ha v e the follo wing diagram y   ? ? ? ? ? ? ? x          a          b   ? ? ? ? ? ? ? c          d   ? ? ? ? ? ? ? The pro ofs of the following Lemmas are similar to the pro ofs of Lemmas 3.8 and 3.9. Lemma 8.5. The multiplic ation is wel l-define d. Lemma 8.6. The multiplic ation is asso ciative. Before pro ceeding with the pro of w e insert a lemma which we shall use often. Lemma 8.7. If [ a, b ] ∈ G , then [ xa, xb ] = [ a, b ] for some x ∈ C such that xa is define d in C . Pr o of. Let [ a, b ] ∈ G where xa is defined in C for some x ∈ C . It is clear that [ xa, xb ] ∈ G . Since C is righ t rev ersible and co d ( xa ) = co d ( a ) we hav e that txa = r a for some t, r ∈ C , by the right rev ersiblilit y of C . By cancellativit y in C we get tx = r . Since dom ( a ) = dom ( b ), it follows that r b is defined. Hence txb = r b . Thus [ a, b ] = [ xa, xb ].  F rom Lemma 3.11, we can easily deduce the follo wing lemma. Lemma 8.8. The identities of G have the form [ a, a ] wher e a ∈ C . Supp ose that [ a, b ] ∈ G . By Condition (C), there exists x ∈ C such that xa is defined. Using Lemma 8.7 and the definition of multiplication, we get [ a, a ][ a, b ] = [ xa, xb ] = [ a, b ] . GR OUPOIDS OF LEFT QUOTIENTS 27 Similarly , [ a, b ][ b, b ] = [ y a, y b ] = [ a, b ] , for some y ∈ C suc h that y b is defined. Hence d ([ a, b ]) = [ a, a ] and r ([ a, b ]) = [ b, b ]. By the abov e argumen t and Lemma 8.6, the following lemma is clear. Lemma 8.9. G is a c ate gory. If [ a, b ] ∈ G , then it is clear that [ b, a ] ∈ G . By Condition (C) and Lemma 8.7, [ a, b ][ b, a ] = [ y a, y a ] = [ a, a ] = d ([ a, b ]) for some y ∈ C suc h that y b is defined. Similarly , [ b, a ][ a, b ] = [ b, b ] = r ([ a, b ]). That is, [ b, a ] is the inv erse of [ a, b ] in G . Th us we ha ve Lemma 8.10. G is a gr oup oid. If a ∈ C , b y (C) there exists x ∈ C such that xa is defined. It is clear that [ x, xa ] ∈ G . If ( y , y a ) ∈ e G , then as co d ( xa ) = co d ( y a ) there exist u, v ∈ C such that uxa = v y a . Since C is cancellative w e hav e that ux = v y , that is, [ x, xa ] = [ y , y a ]. Th us w e hav e a well defined map θ : C − → G giv en b y aθ = [ x, xa ] where x ∈ C suc h that xa is defined. Lemma 8.11. The mapping θ is an emb e dding of C in G . Pr o of. T o show that θ is one-to-one, let [ x, xa ] = [ y , y b ] for some x, y ∈ C so that ux = v y and uxa = v y b for some u, v ∈ C . Hence a = b . F or a, b ∈ C suc h that ab is defined. W e hav e aθ bθ = [ x, xa ][ y , y b ] = [ ux, v y b ] = [ ux, uxab ] = ( ab ) θ , where uxa = v y for some u, v ∈ C . Thus θ is a homomorphism.  By Lemma 8.11, w e can regard C as a subsemigroup oid of G . Let [ a, b ] ∈ G and aθ = [ x, xa ], bθ = [ y , y b ] where xa and y b are defined for some x, y ∈ C . As co d ( x ) = dom ( a ) = dom ( b ) = co d ( y ), b y the righ t rev ersibilit y of C there are elemen ts u, v ∈ C with ux = v y . Hence ( aθ ) − 1 ( bθ ) = [ x, xa ] − 1 [ y , y b ] = [ xa, x ][ y , y b ] = [ uxa, v y b ] = [ uxa, uxb ] b y Lemma 8.7 , = [ a, b ] . Hence C is a left order in G . This completes the pro of of Theorem 8.3.  GR OUPOIDS OF LEFT QUOTIENTS 28 The follo wing lemma can b e deduced from Corollary 4.2. Lemma 8.12. If a semigr oup oid C is a left or der in gr oup oids G and P , then G and P ar e isomorphic by an isomorphism which r estricts to the identity map on C . Let C b e a semigroup oid. W e say that it satisfies the c onne cte d c ondition if, for any a, b ∈ C there exist c, d ∈ C with dom ( c ) = dom ( d ) such that co d ( c ) = co d ( a ) and co d ( d ) = co d ( b ). W e conclude this section with the follo wing results. The pro ofs can b e deduced easily from those for categories in Sections 5 and 6. Lemma 8.13. L et S b e a semigr oup. If S is a left I-or der in a primitive inverse (Br andt) semigr oup Q , then C = S ∗ is a left q-or der in the (c onne cte d) gr oup oid Q ∗ . 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