Extending the Ehresmann-Schein-Nambooripad Theorem

EXTENDI NG THE EHRESMANN -SCHEIN-NAMBOOR I P AD THEOREM CHRISTOPHER HOLLINGS Abstract. W e extend the ‘ ∨ -premor phisms’ par t o f the Ehresmann-Schein- Nambo oripad Theorem to the case of tw o-sided r estriction semigroups a nd inductive categorie s, following on fro m a result of Lawson (1991 ) for the ‘mor- phisms’ part. How ev er, it is s o-called ‘ ∧ -premorphisms’ which ha v e prov ed useful in r e c en t years in the study of partia l actions. W e therefor e obtain an Ehresmann-Schein-Nambo oripad- t yp e theor e m for (ordered) ∧ -premorphisms in the c a se of tw o-sided restriction semigr oups and inductive categor ies. As a corolla r y , w e obtain suc h a theorem in the in v erse case. 1. Intro duction In their study of E -unitary co v ers for in v erse semigroups, McAlister and Reilly [17] made the following definitions: Definition 1.1. [17, D efinition 3 .4 ] Let S and T b e in verse sem igroups. A ( ∨ , i ) - pr emorphism is a function θ : S → T such that ( ∨ 1) ( st ) θ ≤ ( sθ )( tθ ). Definition 1.2. [17, D efinition 4 .1 ] Let S and T b e in verse sem igroups. A ( ∧ , i ) - pr emorphism is a function θ : S → T such that 1 ( ∧ 1) ( sθ )( tθ ) ≤ ( st ) θ ; ( ∧ 2) ′ ( sθ ) − 1 = s − 1 θ . (W e note t hat in [17], a ( ∨ , i )-premorphism w as termed a v -pr ehomomorphism , whilst in [16, p. 80], it is termed simply a pr ehomomorphism . In [17], a ( ∧ , i )- premorphism was called a ∧ -pr e h omomorphism ; in [16, p. 80], it is called a dual pr ehomomo rphism .) Date : August 2, 2018. 2000 Mathematics Su bje ct Classific ation. 2 0 M 18 , 20 L 05, 2 0 M 15. Key wor ds and phr ases. r estriction semigroup, inv erse semigroup, inductive category , induc- tive gr o upo id, premorphism. This w ork was completed as pa rt of P r o ject P OCTI/01 43/20 07 of CA UL, financed by F CT and FEDER, and als o as pa rt of F CT p ost-do ctor al r esearch gra nt SFRH/BP D/3469 8/200 7. Thanks m ust go to bo th Ma rk La wson and Victoria Gould for a n um ber of useful comments. 1 W e place a ′ on condition ( ∧ 2) ′ , as we will sho rtly r eplace this co ndition by a weaker one and w e wish to reserve the lab el ( ∧ 2) for that. 1 2 CHRISTOPHER HO LLINGS These functions were cen tra l to McAlister a nd Reilly’s constructions: see their Theorems 3.9 and 4.5 [17]. In the presen t pap er, w e will study generalisations of the functions of D efini- tions 1.1 and 1.2 in the case o f so-called two - s ide d r estriction semigr oups (un til recen tly , termed we akly E -ample semigr oups ). These are an extremely na t ur a l class o f semigroups which generalise in v erse semigroups a nd whic h ar ise from partial transforma t io n monoids in a manner analogous to the w a y in whic h in- v erse semigroups arise from symmetric in vers e monoids. The concrete description of suc h a semigroup is a s follo ws. Let P T X denote the collection of all partial mappings of a set X , i.e., all mappings A → B , where A, B ⊆ X . W e comp ose elemen ts of P T X (from left to right) according to the usual rule for comp osition of partial mappings, namely , that emplo y ed in the symmetric inv erse monoid I X . Under this comp osition, P T X clearly forms a monoid, whic h w e term the p artial tr ansformation mo n oid of X . Similarly , P T ∗ X , t he dual p artial tr ansformation monoid of X , is the collection of all partial mappings of X with comp o sition p erformed fr om right to left . W e denote by I A the partia l identit y mapping on a subset A ⊆ X ; the collection E X of all suc h partial iden tities forms a subsemi- lattice of b oth P T X and P T ∗ X . W e no w consider the unary op erat io n on part ia l transformations whic h is g iv en by α 7→ I dom α . In P T X , w e denote this op eration b y + ; in P T ∗ X , w e denote it by ∗ . L et S b e a semigroup. W e call S a two-side d r estriction semigr oup if (1) S is isomorphic to a subsemigroup of some P T X that is closed under + (via a n isomorphism φ ); (2) S is isomorphic to a subsemigroup of some P T ∗ Y that is closed under ∗ (via a n isomorphism ψ ); (3) the semilattices { ( sφ ) + : s ∈ S } and { ( sψ ) ∗ : s ∈ S } are isomorphic. Suc h semigroups ha v e app eared in a r ange o f conte xts (see [10]) a nd hav e an alternativ e, abstract c haracterisation whic h will b e used t hroughout this pap er (see Section 2). In extending the functions of Definitions 1.1 and 1.2 to t he case of t wo-sided re- striction semigroups, our particular in terest is in obtaining a generalisation of the celebrated Ehr esm ann-Schein-Namb o orip ad Th e or em (hereafter, ESN Th e or em ). This theorem establishes a fundamental connection b et w een inv erse semigroups and inductiv e gr o up oids. The formal statemen t of the ESN Theorem (a s it ap- p ears in [16], for whic h b o ok it pro vides the main fo cus) is as f o llo ws: Theorem 1.3. [16, Theorem 4 .1.8] The c ate gory of in v e rse semigr oups and ( ∨ , i ) - pr emorphism s is isomorphic to the c ate gory of inductive gr oup oids and or der e d functors; the c ate gory of inverse semigr oups and morphisms is isomorphic to the c ate gory of inductive gr o up oids and inductive functors. (An o r der e d functor is simply an order-preserving functor, whilst a inductive functor is an ordered functor whic h also preserv es t he ‘meet’ op eration in an inductiv e gro up oid.) EXTENDING THE E SN THEOREM 3 The ‘mor phisms’ part of this result has already b een extended to the case of t w o-sided restriction semigroups b y La wson [15]; in Lawson’s result, induc- tiv e group o ids are replaced b y inductiv e categor ies and morphisms b y ( 2 ,1,1)- morphisms. W e will complete the generalisation b y considering t he ‘( ∨ , i )-pre- morphisms’ part. F urthermore, w e will obtain a version of this theorem f o r ( ∧ , i )- premorphisms, via the more general case of t w o- sided restriction semigroups; it is this ty p e of premorphism whic h has prov ed most useful in recen t y ears in the study of partial actions. W e note that arbitrar y ( ∧ , i )- premorphisms do not com- p ose to give another ( ∧ , i )-premorphism. This problem is solv ed (b oth in the in v erse case, and in the theory we will dev elop for restriction semigroups) if we also insist that the f unctions b e o rder-preserving. The structure of the pap er is as follow s. W e b egin with Section 2, in whic h w e record some results on t w o - sided restriction semigroups whic h will b e o f use in later sections. F urt her preliminary definitions and results fo llow in Section 3: in t he first part of the section, w e in tro duce v arious category-theoretic notions, including that of an inductiv e categor y; in the second part, we define the Szen- dr ei exp ansion of an inductiv e gr oup oid — a piece of a lg ebraic machine ry whic h will b e needed briefly in Section 6 . In Section 4 w e define the notion o f ( ∨ , r ) - pr emorphism s f o r tw o-sided restriction semigroups and prov e a new v ersion (The- orem 4.1) of the ESN Theorem with these functions as the arrows of a cate- gory of tw o-sided restriction semigroups. W e show that the existing result in the inv erse case f o llo ws from ours as a corollary . In Section 5, w e mov e onto the ‘ ∧ -premorphisms’ part of the pap er. By a dapting concepts encoun tered in the study of pa rtial actions, we prov e y et a no ther v ersion (Theorem 5 .1) of the ESN Theorem, this time for tw o-sided restriction semigroups and or der e d ( ∧ , r ) - pr emorphism s . In f act, we will need t o prov e t w o versions of this theorem: it will turn out that the most naive notion o f ‘( ∧ , r )-premorphism’, i.e., that obtained b y replacing condition ( ∧ 2) ′ in Definition 1.2 b y a condition in v olving + and ∗ , will not suffice for our purp o ses and we will therefore also need the notion of a str ong ( ∧ , r )-premorphism (see Theorem 5.9). A v ersion of the ESN Theorem fo r in v erse semigroups and ordered ( ∧ , i )-premorphisms (Theorem 6.1) will follow in Section 6, as a corollary to t he results of Section 5, thanks to our intro duction o f strong ( ∧ , r )-premorphisms. 2. Restriction semigroups In this section, we summarise the p ertinen t details of the theory of tw o-sided restriction semigroups. The mat erial of this section app ears in a range of pub- lished sources (see, for example, [4, 5, 8, 13, 15]). How ev er, there are only t w o places where man y of the relev an t definitions and results hav e b een collat ed into a single resource: the notes [9] and Chapter 2 of the author’s Ph.D. thesis [11]. The reader is referred to these sources for further details and fo r more extensiv e 4 CHRISTOPHER HO LLINGS references. F or a more easily accessible published source, see Section 1 of [12], written for monoids. Notice from our commen ts in the Introduction that w e can easily define one- sided v ersions of these semigroups: left r e s triction semigr oups (subsemigroups of P T X closed under + ) and right r estriction semigr oups (subsemigroups of P T ∗ Y closed under ∗ ). Ho w eve r, in this pa p er, w e will only b e interes ted in the tw o-sided v ersion. F rom here on, w e therefore drop the qualifier ‘tw o-sided’; henceforth, whenev er we use the term ‘restriction semigroup’, it can b e tak en to mean the tw o- sided v ersion. W e also no t e here that left/ r igh t/t w o- sided restriction semigroups ha v e app eared in a range of contexts under a num b er of differen t names: see [10] for further r eferences. The term ‘restriction semigroup’ is a recen t attempt to harmonise terminology a nd originates with [3]. In the In tro duction, w e saw the concrete characterisation of restriction semi- groups as subsemigroups of partial transformation monoids. W e now giv e an abstract description. Let S b e a semigroup and supp ose that E ⊆ E ( S ) is a subsemilattice of S . W e define the (equiv alence) relations e R E and e L E on S , with resp ect to E , b y the rules that a e R E b ⇐ ⇒ ∀ e ∈ E [ ea = a ⇔ eb = b ]; a e L E b ⇐ ⇒ ∀ e ∈ E [ ae = a ⇔ be = b ] , for a, b ∈ S . Thus , t w o elemen ts a, b ar e e R E -related if, and only if, they hav e the same left identities in E . Similarly , a e L E b if, a nd only if, a and b hav e the same righ t identities in E . The abstract definition of a r estriction semigroup runs as follo ws: Definition 2.1. A semigroup S with subsemilattice E ⊆ E ( S ) is (two-s i d e d) r estriction semigr oup (with r esp e ct to E ) if (1) ev ery elemen t a is b oth e R E - and e L E -related to an idemp o ten t in E , de- noted a + and a ∗ , resp ectiv ely; (2) e R E is a left congruence, whilst e L E is a righ t congruence; (3) for all a ∈ S and all e ∈ E , ae = ( ae ) + a and ea = a ( ea ) ∗ . Th us, in a restriction semigroup S , a e R E b ⇔ a + = b + and a e L E ⇔ a ∗ = b ∗ . The idempo ten ts a + and a ∗ are left and right iden tities for a , respective ly . W e note that a + and a ∗ are necessarily unique. It is also clear that if e ∈ E , then e + = e = e ∗ . If we w ere to consider only the parts of D efinition 2.1 whic h relate to e R E and + (resp ectiv ely , e L E and ∗ ), then we w ould hav e a left (resp ectiv ely , right ) restric tion semigroup. EXTENDING THE E SN THEOREM 5 Using [9, Theorem 6.2], it is p ossible to connect the concrete and abstract ap- proac hes to restriction semigroups b y showin g that left (righ t) restriction semi- groups a re, up to isomorphism, pr e cisely (2,1 ) - subalgebras of ( dua l) partial tr ans- formation monoids. Ho w eve r, tw o-sided restriction semigroups cannot b e re- garded as (2,1,1 )-subalgebras o f par t ial transformation monoids. Note tha t R ⊆ e R E and L ⊆ e L E , fo r any E . It is easy to see that, in a regular semigroup, R = e R E ( S ) and L = e L E ( S ) . It follows that restriction semigroups generalise inv erse semigroups, since ev ery in v erse semigroup is a restriction semi- group with a + = aa − 1 and a ∗ = a − 1 a . Restriction semigroups also g eneralise the ample (formerly , type-A) semigroups of F o un tain [4, 5]. W e no te a pair of useful iden tities whic h follow easily f r om conditio n (2 ) of Definition 2.1: Lemma 2.2. [5, Prop osition 1.6(2 )] L et S b e a r e s triction sem igr oup, for some E ⊆ E ( S ) , and let s, t ∈ S . Then ( st ) + = ( st + ) + and ( st ) ∗ = ( s ∗ t ) ∗ . Both P T X and P T ∗ X p ossess an obvious natural partial or der (i.e., a partial order whic h is compatible with m ultiplication and whic h restricts to the usual partial order on idemp o t ents), defined b y α ≤ β ⇐ ⇒ α = β | dom α . (2.1) In the abstract characteris ation of a restriction semigroup S , the ordering of (2.1) b ecomes the following natura l part ia l order a ≤ b ⇐ ⇒ a = eb ⇐ ⇒ a = bf , (2.2) for some idemp oten ts e, f ∈ E . Equiv alen tly , a ≤ b ⇐ ⇒ a = a + b ⇐ ⇒ a = ba ∗ . (2.3) This equiv alence is j ustified (f o r the ‘ + ’ par t ) in [12, § 1]. 3. Fur the r p reliminaries In this section, w e describ e the relev ant existing results connecting restriction semigroups and inductiv e categories, and in tro duce some algebraic machine ry for later use. 3.1. Categories. W e b egin by giving an explicit definition of an arbitra ry cate- gory . The results quoted in this subsection will take care of the ‘ob jects’ parts of our main results: the up coming ESN-type Theorems 4.1, 5.1, 5.9 and 6.1. Let C b e a class and let · b e a partial binary o p eration o n C . F or x, y ∈ C , w e will write ‘ ∃ x · y ’ to mean ‘the pro duct x · y is defined’. Whenev er we write ‘ ∃ ( x · y ) · z ’, it will b e understo o d that we mean ∃ x · y and ∃ ( x · y ) · z . An elemen t e ∈ C is idemp otent if ∃ e · e and e · e = e . The ide ntities (o r obje cts ) of C are those idemp oten ts e whic h satisfy: [ ∃ e · x ⇒ e · x = x ] and [ ∃ x · e ⇒ x · e = x ]. W e denote the subset of iden t it ies of C by C o (‘ o ’ for ‘ob jects’). 6 CHRISTOPHER HO LLINGS Definition 3.1. Let C b e a class and let · b e a partial binary op eration on C . The pair ( C , · ) is a c ate gory if the follo wing conditions hold: (Ca1) ∃ x · ( y · z ) ⇐ ⇒ ∃ ( x · y ) · z , in whic h case x · ( y · z ) = ( x · y ) · z ; (Ca2) ∃ x · ( y · z ) ⇐ ⇒ ∃ x · y and ∃ y · z ; (Ca3) for eac h x ∈ C , there exist unique iden tities d ( x ) , r ( x ) ∈ C o suc h that ∃ d ( x ) · x and ∃ x · r ( x ). The iden tity d ( x ) is called the domain of x and r ( x ) is the r ange o f x . If C is simply a set, then w e call ( C , · ) a smal l category . A gr oup oid is a small catego r y in whic h the following additional condition holds: (G) f o r eac h x ∈ G , there is an x − 1 ∈ G suc h that ∃ x · x − 1 and ∃ x − 1 · x , with x · x − 1 = d ( x ) and x − 1 · x = r ( x ). This is essen tially the definition of [16, p. 78], but with domain and range switc hed, since we will b e comp o sing functions from left to right. W e note that if ( C , · ) is a category a nd x, y ∈ C , then ∃ x · y ⇐ ⇒ r ( x ) = d ( y ) . Note further tha t if ( C , · ) has precisely one ob ject e , then all pro ducts are neces- sarily defined and it f o llo ws that ( C , · ) is , in fact, a monoid with iden tit y e . Th us, a category ma y b e regarded as a generalisation of a monoid. Before pro ceeding further, w e mak e the imp orta n t observ ation that in the r e- mainder of this pap er, the word ‘category’ will b e used in tw o sligh tly differen t, though equiv alent, senses. First, we w ill ha v e ‘catego r ies’ as generalised monoids, in t he sense of D efinition 3.1; second, w e will hav e ‘catego ries’ in the more tra- ditional ‘ob jects’ and ‘morphisms’ sense (as in, for example, [14, Definition 1.1]). All categories considered in the first sense will b e small catego r ies. Thus , for example, the ‘inductiv e catego ries’ of Theorem 4.1 are (small) categories in the sense of D efinition 3.1, but the category of inductiv e categories (and tha t of re- striction semigroups) is rega rded as a category in the traditiona l ‘ob jects’ and ‘morphisms’ sense. W e no w in tro duce an ordering on a category C : Definition 3.2. Let ( C , · ) b e a category (in the sense of Definition 3.1) and let C b e partially or dered by ≤ . The triple ( C , · , ≤ ) is a n or der e d catego ry if the follo wing conditions hold: (Or1) a ≤ c , b ≤ d , ∃ a · b a nd ∃ c · d = ⇒ a · b ≤ c · d ; (Or2) a ≤ b = ⇒ r ( a ) ≤ r ( b ) and d ( a ) ≤ d ( b ); (Or3) (i) f ∈ C o , a ∈ C , f ≤ r ( a ) = ⇒ there exists a unique elemen t, denoted a | f , suc h that a | f ≤ a and r ( a | f ) = f ; (ii) f ∈ C o , a ∈ C , f ≤ d ( a ) = ⇒ there exists a unique elemen t, denoted f | a , suc h that f | a ≤ a and d ( f | a ) = f . An or der e d gr oup oid is a small ordered categor y in whic h condition (G) holds. EXTENDING THE E SN THEOREM 7 The elemen t a | f of condition (Or3)( i) is called the c or estriction (of a to f ), whilst the elemen t f | a of condition (Or 3 )(ii) is called the r estriction (of f to a ). W e note that, for e, f ∈ C o , there is no am biguit y in the no t ation ‘ e | f ’: the restriction e | f and the corestriction e | f coincide whenev er b oth a re defined. Note further that in this case we ha v e e = f = e | f , b y definition of restrictions and corestrictions. It is clear that an y function whic h resp ects domains, ranges and ordering (for example, an ordered functor) will also resp ect restrictions and corestrictions. W e r ecord the follow ing prop erties of the restriction and corestriction in a n ordered category: Lemma 3.3. L et ( C , · , ≤ ) b e a n or der e d c ate gory. L et a ∈ C an d e, f ∈ C o with f ≤ e ≤ r ( a ) . Then ( a | e ) | f = a | f . Conse quently, a | f ≤ a | e . Dual ly, if a ∈ C and e, f ∈ C o with f ≤ e ≤ d ( a ) , then f | ( e | a ) = f | a , henc e f | a ≤ e | a . (Our Lemma 3.3 is essen tially L emma 4.4 o f [1 5].) Note that in an inductiv e gro up oid G , corestrictions may b e defined in terms of restrictions: for a ∈ G and f ∈ G o with f ≤ r ( a ), the corestriction a | f is giv en b y a | f = ( f | a − 1 ) − 1 [7, p. 178]. Lemma 3.4. L et ( C , · , ≤ ) b e a n o r der e d c ate gory an d le t a, b ∈ C . I f a ≤ b , then a = d ( a ) | b = a | r ( b ) . In p articular, a = d ( a ) | a = a | r ( a ) . (This result app ears in [1 ] f o r inductiv e cancellativ e categories; the pro of carries o v er the presen t case without mo dificatio n.) In an ordered catego ry ( C , · , ≤ ), if the greatest low er b ound of e, f ∈ C o exists, then w e denote it by e ∧ f . Definition 3.5. An inductive category ( C , · , ≤ ) is an ordered category in whic h the follow ing additional condition holds: (I) e, f ∈ C o = ⇒ e ∧ f exists in C o . An inductive gr oup oid is a small inductiv e category in whic h condition ( G) holds. The foregoing sequence of definitions has b een building to the follow ing result, whic h app ears in [15, § 5 ]: Theorem 3.6. L et S b e a r estriction semigr oup with r esp e ct to som e semilattic e E , and let S have natur al p artial or de r ≤ . If we de fine the restricted pro duct · in S by a · b = ( ab if a ∗ = b + ; undefine d otherwise , then ( S, · , ≤ ) is an in d uctive c ate gory with S o = E , d ( x ) = x + and r ( x ) = x ∗ . The restriction f | a in the inductiv e category ( S, · , ≤ ) is simply the pro duct f a in the restriction semigroup S , since f a ≤ a and ( f a ) + = ( f a + ) + = f , b y Lemma 2.2, as f ≤ a + . Similarly , a | f = af and e ∧ f = ef . 8 CHRISTOPHER HO LLINGS W e no w define the pseudopr o duct ⊗ in an inductiv e category ( C , · , ≤ ) b y 2 a ⊗ b = [ a | r ( a ) ∧ d ( b )] · [ r ( a ) ∧ d ( b ) | b ] . The pseudopro duct is eve rywhere-defined in C (t ha nks to (I)) and coincides with the pro duct · in C whenev er · is defined. T o see this, recall that if ∃ a · b , then r ( a ) = d ( b ), so that a ⊗ b = [ a | r ( a )] · [ d ( b ) | b ]. Then a ⊗ b = a · b , b y Lemma 3.4 . W e record the following prop erties of the pseudopro duct for later use: Lemma 3.7. L et ( C , · , ≤ ) b e an inductive c ate gory and let a ∈ C , e ∈ C o . Then e ⊗ a = e ∧ d ( a ) | a and a ⊗ e = a | r ( a ) ∧ e. (This result app ears in [1 ] f o r inductiv e cancellativ e categories; the pro of carries o v er the presen t case without mo dificatio n.) Theorem 3.8. [1 5, § 5] I f ( C , · , ≤ ) is an inductive c ate gory, then ( C , ⊗ ) is a r estriction semigr oup with r esp e ct to C o . Let S b e a restriction semigroup. W e will denote the inductiv e category asso- ciated to S by C ( S ). Similarly , if C is an inductiv e category , then w e will denote its asso ciated restriction semigroup by S ( C ). The fo llowing result is implicit in [15, § 5]: Theorem 3.9. L et S b e a r estriction semigr oup and C b e an inductive c ate gory. Then S ( C ( S )) = S and C ( S ( C )) = C . Theorems 3.6, 3.8 and 3.9 prov e the equ iv alence of the ob j ects of the categories in the up coming Theorems 4.1 , 5.1 and 5.9. It o nly remains f or us to deal with the arrow s. W e conclude this subsection with a prop erty of ordered categories whic h will b e immensely useful later in the pap er. Lemma 3.10. L et ( C , · , ≤ ) b e an or der e d c ate gory and let a, b ∈ C . If a, b ≤ c , for some c ∈ C , and either d ( a ) = d ( b ) or r ( a ) = r ( b ) , then a = b . (This result app ears in [1 ] f o r inductiv e cancellativ e categories; the pro of carries o v er the presen t case without mo dificatio n.) 3.2. The Szendrei expansion of an inductiv e group oid. W e tak e this o p- p ortunity to in tro duce a n algebraic to ol whic h w e will require briefly in Section 6: the Szendr ei exp ansion of an inductiv e group oid, as defined by Gilb ert [7]. The concept of a ‘semigroup expansion’ w as first in tro duced b y Birg et and Rho des in [2], and is simply a sp ecial ty p e of functor from one category of semi- groups to another. Amongst the v ario us expansions in the literat ure are a num b er of differen t vers ions o f the so-called Szendr ei exp an sion . The original Szendrei 2 In the interests of reducing the num b er o f br ack ets, expressio ns such as a | r ( a ) ∧ d ( b ) will be understo o d to mean a | ( r ( a ) ∧ d ( b )); of cours e, the alternative, ( a | r ( a )) ∧ d ( b ), makes no sense if a is not an iden tit y . EXTENDING THE E SN THEOREM 9 expansion was in tro duced in [18] and w as there applied to groups; it was subse- quen tly extended to monoids in [6 ]. Gilb ert [7 ] ha s studied the partial actions o f inductiv e group oids as a w a y of informing the study of pa r t ial actions of inv erse semigroups. In the course of t his w ork, he has defined a Szendrei expansion for inductiv e group oids. Let G b e an inductiv e gro up oid. F or each iden tit y e ∈ G o , we define the set star e ( G ) b y star e ( G ) = { g ∈ G : d ( g ) = e } . Let F e (star e ( G )) b e t he collection of all finite subsets of star e ( G ) whic h con tain e , and put F ∗ ( G ) = [ e ∈ G o F e (star e ( G )) . W e no w mak e the following definition: Definition 3.11. [7, p. 179] Let G b e an inductiv e group oid. The S zendr ei exp ansion 3 of G is the set Sz( G ) = { ( U, u ) ∈ F ∗ ( G ) × G : u ∈ U } , together with the op eration ( U, u )( V , v ) = ( ( U, uv ) if r ( u ) = d ( v ) and U = uV ; undefined o t herwise . W e note that Sz( G ) has iden tities Sz( G ) o = { ( E , e ) ∈ Sz( G ) : e ∈ G o } . W e note also that G ilb ert’s definition applies, more generally , to or der e d group- oids, but w e will only concern ourselv es with the inductiv e case. Prop osition 3.12. [7, P rop osition 3.1 & Corollary 3 .3 ] If G is an in d uctive gr oup oid, then Sz( G ) is an inductive gr oup oid wi th ( U, u ) − 1 = ( u − 1 U, u − 1 ) , d (( U, u )) = ( U, d ( u )) , r (( U, u )) = ( u − 1 U, r ( u )) , (3.1) and or dering ( U, u ) ≤ ( V , v ) ⇐ ⇒ u ≤ v in G and d ( u ) | V ⊆ U, wher e d ( u ) | V = { d ( u ) | w : w ∈ V } . F or ( E , e ) ∈ Sz( G ) o with ( E , e ) ≤ d (( A, a )) , the r estriction 4 is given by ( E , e ) | ( A, a ) = ( E , e | u ) , whilst, for any ( E , e ) , ( F , f ) ∈ Sz( G ) o , we have ( E , e ) ∧ ( F , f ) = (( e ∧ f ) | ( E ∪ F ) , e ∧ f ) . (3.2) 3 Gilber t r efers to this as the Bir get-R ho des ex p ansion but we adopt the term Szendr ei ex- p ansion for consistency with previo us definitio ns. 4 W e need not give the corestrictio n explicitly , thanks to the co mmen ts following Lemma 3.3. 10 CHRISTOPHER HO LLINGS The pseudopro duct in Sz( G ) may b e written: ( U, u ) ⊗ ( V , v ) = ( d ( u | r ( u ) ∧ d ( v )) | U ∪ u ⊗ V , u ⊗ v ) , (3.3) where u ⊗ V = { u ⊗ w : w ∈ V } [7, Theorem 3.2]. Note also tha t w e can inject an y inductiv e group o id G in to Sz( G ) via the mapping ι : G → Sz( G ), giv en by g ι = ( { d ( g ) , g } , g ). The Szendrei expansion of an induc tiv e group o id will be used to pro v e Lemma 6.5. 4. ( ∨ , r ) - premorphisms The goal of this section is the pro of of the following theorem, and its connection with the ‘( ∨ , i )-premorphisms’ part of Theorem 1.3 . Theorem 4.1. The c ate gory of r estriction semigr oups and ( ∨ , r ) -p r emorphisms is isomorphic to the c ate gory of inductive c ate gories and or de r e d functors. W e not e that the ‘ob jects’ part of this theorem ha s b een taken care of by the results of Section 3; it remains to deal with the ‘arrows ’ part. W e b egin by defining the notio n of a ( ∨ , r ) -pr emo rphism . Definition 4.2. Let S a nd T b e restriction semigroups. A ( ∨ , r ) - p r emorphism is a function θ : S → T suc h that ( ∨ 1) ( st ) θ ≤ ( sθ )( tθ ); ( ∨ 2) s + θ ≤ ( sθ ) + and s ∗ θ ≤ ( sθ ) ∗ . The notion of a ( ∨ , r )- premorphism generalises that of a ( ∨ , i )-premorphism. T o see this, we m ust first record the follo wing concerning ( ∨ , i )-premorphisms: Lemma 4.3. [16, Theorem 3.1.5] L et θ : S → T b e a ( ∨ , i ) -pr emorphism. Th e n θ r esp e cts inverses and the natur al p artial or der. W e no w ha v e: Lemma 4.4. L et θ : S → T b e a function b etwe en inverse se migr oups. Then θ is a ( ∨ , i ) -pr emorp h ism i f , and on ly if, it is a ( ∨ , r ) -pr emorphism . Pr o of. ( ⇐ ) Immediate. ( ⇒ ) Supp ose that θ : S → T is a ( ∨ , i )-premorphism. Then s + θ = ( s s − 1 ) θ ≤ ( sθ )( s − 1 θ ) = ( sθ )( sθ ) − 1 = ( sθ ) + , using Lemma 4.3. Similarly , s ∗ θ ≤ ( sθ ) ∗ .  W e note some useful prop erties of ( ∨ , r )- premorphisms: Lemma 4.5. L et S a nd T b e r estriction semigr oups w i th r esp e ct to semilattic es E and F , r esp e ctively. If θ : S → T is a ( ∨ , r ) -pr emorphism, then (a) e ∈ E ( S ) ⇒ eθ ∈ E ( T ) ; (b) e ∈ E ⇒ eθ ∈ F ; (c) ( sθ ) + = s + θ and ( sθ ) ∗ = s ∗ θ ; EXTENDING THE E SN THEOREM 11 (d) θ is or der-pr ese rving. Pr o of. (a) Let e ∈ E ( S ). F rom ( ∨ 1), eθ = e 2 θ ≤ ( eθ ) 2 . Then, b y (2.3), eθ = ( eθ ) + ( eθ ) 2 = ( eθ ) 2 , hence eθ ∈ E ( T ). (b) Let e ∈ E . F ro m ( ∨ 2), eθ = e + θ ≤ ( eθ ) + . Again by (2.3), eθ = ( eθ ) + ( eθ ) + = ( eθ ) + , hence eθ ∈ F . (c) W e deal with the ‘ + ’ part; the ‘ ∗ ’ part is s imilar. W e will sho w that ( sθ ) + ≤ s + θ . The desired result will then follo w by com bining this with ( ∨ 2). Let s ∈ S . F rom ( ∨ 1), sθ = ( s + s ) θ ≤ ( s + θ )( sθ ), so sθ = ( s + θ )( sθ )( sθ ) ∗ = ( s + θ )( sθ ) , (4.1) once again b y ( 2 .3). Applying + to b oth sides o f (4.1) g ives ( sθ ) + = (( s + θ )( sθ )) + = (( s + θ )( sθ ) + ) + = ( s + θ )( sθ ) + , using Lemma 2.2, and since s + θ ∈ F , b y (b). Hence ( sθ ) + ≤ s + θ . (d) Supp ose that s ≤ t in S . Then s = s + t , b y (2.3), so sθ = ( s + t ) θ ≤ ( s + θ )( tθ ) ≤ tθ in T .  Using Lemma 4.5(d), it is easily v erified that the comp osition of tw o ( ∨ , r )- premorphisms is a ( ∨ , r )- pr emorphism, hence restriction semigroups and ( ∨ , r )- premorphisms constitute a cat ego ry . W e are now ready to prov e the ‘arrows’ part of Theorem 4.1, whic h we will break do wn in to tw o parts (Prop o sitions 4.8 and 4.9). Before we do so, ho w ev er, we first record the follo wing, whic h will b e used a n um ber of times in t he remainder o f t his pap er: Lemma 4.6. L et α : S → T b e an or der-pr eserving function o f r estriction semi- gr oups. We define C ( α ) : C ( S ) → C ( T ) to b e the same function on the underly- ing sets. Then C ( α ) is or d e r-pr eserving. L et β : C → D b e an or der-pr eserving function o f inductive c ate gories. We define S ( β ) : S ( C ) → S ( D ) to b e the same function on the underlying sets. Th e n S ( β ) is or der-pr es e rving. Pr o of. Let s, t ∈ C ( S ). Then s ≤ t in C ( S ) ⇒ s ≤ t in S ⇒ sα ≤ tα in T ⇒ s C ( α ) ≤ t C ( α ) in C ( T ) . The pro of of the second par t is similar.  W e no w also giv e a forma l definition for the o rdered functors whic h app ear in Theorem 4.1: Definition 4.7. Let φ : C → D b e a function b et w een ordered categories C and D . W e call φ an or de r e d func tor if (1) ∃ x · y in C ⇒ ∃ ( xφ ) · ( y φ ) in D and ( xφ ) · ( y φ ) = ( x · y ) φ ; (2) x ≤ y in C ⇒ xφ ≤ y φ in D . 12 CHRISTOPHER HO LLINGS Prop osition 4.8. L et S and T b e r estriction semig r oups with r e s p e ct to sem i l a t- tic es E and F , r esp e ctively. L et θ : S → T b e a ( ∨ , r ) -pr emo rphism. We define Θ := C ( θ ) : C ( S ) → C ( T ) to b e the same function on the underlying s e ts. Then Θ is an or der e d functor with r esp e ct to the r estricte d pr o ducts in C ( S ) and C ( T ) . Pr o of. Since θ maps E in to F , plus C ( S ) o = E a nd C ( T ) o = F , w e see that Θ maps iden tities in C ( S ) to iden tities in C ( T ). W e now sho w that Θ resp ects restricted pro ducts. Suppose that ∃ s · t in C ( S ). Then s ∗ = t + in S , hence ( s Θ) ∗ = ( sθ ) ∗ = s ∗ θ = t + θ = ( tθ ) + = ( t Θ) + , b y Lemma 4.5( c). W e conclude that ∃ ( s Θ) · ( t Θ) in C ( T ). W e will now use Lemma 3.10 to sho w that ( s · t )Θ = ( s Θ) · ( t Θ). Note first o f all that ( s · t )Θ = ( st ) θ ≤ ( sθ )( tθ ) a nd that ( s Θ) · ( t Θ) = ( sθ )( tθ ) ≤ ( sθ ) ( tθ ). Now , using Lemma 2.2, r (( s · t )Θ) = ( st ) θ ∗ = ( st ) ∗ θ = ( s ∗ t ) ∗ θ = t ∗ θ , since s ∗ = t + . Also, r (( s Θ) · ( t Θ)) = (( sθ )( tθ )) ∗ = (( sθ ) ∗ ( tθ )) ∗ = (( s ∗ θ )( tθ )) ∗ = (( t + θ )( tθ )) ∗ = (( tθ ) + ( tθ )) ∗ = ( tθ ) ∗ = t ∗ θ = r (( s · t )Θ) . Hence, b y L emma 3.10, ( s · t )Θ = ( s Θ) · ( t Θ). Finally , Θ is order-preserving, b y Lemma 4.6.  Prop osition 4.9. L et φ : C → D b e an or der e d functor of ind uctive c ate gories. We d e fine Φ := S ( φ ) : S ( C ) → S ( D ) to b e the same function on the underlying sets. Then Φ is a ( ∨ , r ) -p r emorphism wi th r esp e ct to the pseudopr o ducts in S ( C ) and S ( D ) . Pr o of. Let s, t ∈ S ( C ). Note that if e = s ∗ ⊗ t + , then ( s ⊗ e ) ∗ = ( s ⊗ s ∗ ⊗ t + ) ∗ = ( s ⊗ t + ) ∗ = ( s ∗ ⊗ t + ) ∗ = s ∗ ⊗ t + = ( s ∗ ⊗ t + ) + = ( s ∗ ⊗ t ) + = ( s ∗ ⊗ t + ⊗ t ) + = ( e ⊗ t ) + , so ∃ ( s ⊗ e ) · ( e ⊗ t ) and ( s ⊗ e ) · ( e ⊗ t ) = s ⊗ t . Since φ is a functor, we hav e ( s ⊗ t ) φ = ( s ⊗ e ) φ · ( e ⊗ t ) φ = ( s ⊗ e ) φ ⊗ ( e ⊗ t ) φ, using the fact that · and ⊗ coincide whenev er · is defined. Now, s ⊗ e ≤ s a nd e ⊗ t ≤ t , so ( s ⊗ e ) φ ≤ sφ and ( e ⊗ t ) φ ≤ tφ . Hence ( s ⊗ t )Φ = ( s ⊗ t ) φ = ( s ⊗ e ) φ ⊗ ( e ⊗ t ) φ ≤ ( sφ ) ⊗ ( tφ ) = ( s Φ) ⊗ ( t Φ) . Let s ∈ S . Since functors preserv e domains, w e ha ve ( s Φ) + = d ( sφ ) = d ( s ) φ = s + Φ. Then, in particular, ( s Φ) + ≥ s + Φ. Similarly , ( s Φ) ∗ ≥ s ∗ Φ.  It is clear that if θ : S → T is a ( ∨ , r )-premorphism and φ : C → D is a n ordered functor, then S ( C ( θ )) = θ and C ( S ( φ )) = φ . F urthermore, if θ ′ : T → T ′ is another ( ∨ , r )-premorphism of restriction semigroups, and φ ′ : D → D ′ is another ordered functor of inductiv e categories, then C ( θ θ ′ ) = C ( θ ) C ( θ ′ ) and S ( φφ ′ ) = S ( φ ) S ( φ ′ ). W e ha v e therefore pro v ed Theorem 4.1. EXTENDING THE E SN THEOREM 13 The constructions of Theorems 3 .6 and 3.8 carry ov er to the inv erse case in suc h a w a y that if S is an in v erse semigroup and G is an inductiv e group oid, then C ( S ) is an inductiv e group oid and S ( G ) is an inv erse semigroup (see [16]) . Then, using Lemma 4.4, w e see that t he ‘ ∨ -premorphisms’ part of Theorem 1.3 follo ws f rom Theorem 4.1 as a coro llary . 5. ( ∧ , r ) - premorphisms W e now turn our attention to the deriv ation o f an ESN-t ype Theorem for ‘ ∧ - premorphisms’. Tw o results of this t yp e will b e prov ed in this section. The first will emplo y the more naive vers ion o f a ‘( ∧ , r )-premorphism’, alluded t o in the In tro duction. How ev er, w e will not b e able to pro v e an analogue of Lemma 4.4 for these ( ∧ , r )-premorphisms. In order to mak e the desired connection with the functions o f Definition 1.2, and t hereb y deduce an ESN-type theorem for in v erse semigroups and ( ∧ , i )-premorphisms, we need the no t io n of a str ong ( ∧ , r )- premorphism. In the second par t of this section, we will prov e an ESN-type theorem for strong ( ∧ , r )-premorphisms whic h will f o llo w from the w eak er v ersion as a corollary . 5.1. The wea k er case: ordered ( ∧ , r )-premorphisms. The goal of this s ub- section is the pro of of the f ollo wing theorem: Theorem 5.1. The c ate gory of r estriction sem igr oups an d or der e d ( ∧ , r ) -p r e- morphisms is isomo rphic to the c ate gory of inductive c ate go ries and inductive c ate gory pr efunctors. W e note that the ‘ob jects’ part of this theorem ha s b een tak en care of b y the results of Section 3; it remains to deal with the ‘arrows’ par t . W e mak e the follow ing definition, ba sed up on the one-sided case in [10 ]: Definition 5.2. A function θ : S → T b etw een restriction semigroups is called a ( ∧ , r ) -p r emorphism if ( ∧ 1) ( sθ )( tθ ) ≤ ( st ) θ ; ( ∧ 2) ( sθ ) + ≤ s + θ and ( sθ ) ∗ ≤ s ∗ θ . If, in addition, θ is o rder-preserving, w e call it an or der e d ( ∧ , r ) -pr emorph ism . W e note that whilst a ( ∨ , r ) - premorphism is automatically order- preserving, this is not the case for a ( ∧ , r )-premorphism: we m ust demand this explicitly . As noted in the In tro duction, arbitrary ( ∧ , r )- premorphisms do not comp ose to giv e a ( ∧ , r )- premorphism. Ho w ev er, it is easily v erified that the comp osition of t w o or der e d ( ∧ , r )- premorphisms is an ordered ( ∧ , r )-premorphism. Restriction semigroups together with ordered ( ∧ , r )-premorphisms therefore form a category . Lemma 5.3. L et S a nd T b e r estriction semigr oups w i th r esp e ct to semilattic es E and F , r esp e c tive l y. L et θ : S → T b e a n or der e d ( ∧ , r ) -pr emorp h ism. If e ∈ E , then eθ ∈ F . 14 CHRISTOPHER HO LLINGS Pr o of. Let e ∈ E . Then, b y ( ∧ 2), ( eθ ) + ≤ e + θ = eθ . By (2.3), w e hav e ( eθ ) + = ( eθ ) + ( eθ ) = eθ ∈ F .  W e m ust no w define a corresp onding function b et w een inductiv e cat ego ries: Definition 5.4. A function ψ : C → D b etw een inductiv e categories is called a n inductive c ate gory pr efunctor if (ICP1) ∃ s · t in C ⇒ ( sψ ) ⊗ ( tψ ) ≤ ( s · t ) ψ ; (ICP2) d ( sψ ) ≤ d ( s ) ψ and r ( sψ ) ≤ r ( s ) ψ ; (ICP3) s ≤ t in C ⇒ sψ ≤ tψ in D ; (ICP4) (a) f or a ∈ C and f ∈ C o , aψ | r ( aψ ) ∧ f ψ ≤ ( a | r ( a ) ∧ f ) ψ ; (b) for a ∈ C and e ∈ C o , eψ ∧ d ( aψ ) | aψ ≤ ( e ∧ d ( a ) | a ) ψ . Note that we can use Lemma 3.7 to rewrite condition (ICP4) in a mo r e compact form: (ICP4) ′ (a) fo r a ∈ C a nd f ∈ C o , aψ ⊗ f ψ ≤ ( a ⊗ f ) ψ ; (b) for a ∈ C and e ∈ C o , eψ ⊗ aψ ≤ ( e ⊗ a ) ψ . Lemma 5.5. L et ψ : C → D b e an ind uctive c ate gory pr efunctor of ind uctive c ate gories . I f e ∈ C o , then eψ ∈ D o . Pr o of. Let e ∈ C o , so that e = e + in S ( C ). By (ICP2), d ( eψ ) ≤ d ( e ) ψ in D , so ( eψ ) + ≤ e + ψ = eψ in S ( D ) . Then, b y definition of ordering in S ( D ), ( eψ ) + = ( eψ ) + ⊗ ( eψ ) = eψ . Th us d ( eψ ) = eψ in D . Hence eψ ∈ D o .  Lemma 5.6. Th e c omp osition of two inductive c ate gory pr efunctors is an induc- tive c ate gory pr efunctor. Pr o of. Let ψ 1 : U → V and ψ 2 : V → W b e inductive category prefunctors of inductiv e categories U , V and W . It is easy to sho w that ψ 1 ψ 2 satifies (ICP2)– (ICP4); (ICP1), how ev er, is a little t r ickier. Let s, t ∈ U a nd supp ose that ∃ s · t , i.e., r ( s ) = d ( t ). Put x = sψ 1 and y = tψ 1 . Then ( sψ 1 ψ 2 ) ⊗ ( tψ 1 ψ 2 ) = ( xψ 2 ) ⊗ ( y ψ 2 ) (5.1) = ( xψ 2 | r ( xψ 2 ) ∧ d ( y ψ 2 )) · ( r ( xψ 2 ) ∧ d ( y ψ 2 ) | y ψ 2 ) = ( xψ 2 | r ( xψ 2 ) ∧ d ( y ψ 2 )) ⊗ ( r ( xψ 2 ) ∧ d ( y ψ 2 ) | y ψ 2 ) (since · and ⊗ coincide whenev er · is defined) ≤ ( xψ 2 | r ( xψ 2 ) ∧ d ( y ) ψ 2 ) ⊗ ( r ( x ) ψ 2 ∧ d ( y ψ 2 ) | y ψ 2 ) (b y Lemma 3.3, since r ( xψ 2 ) ∧ d ( y ψ 2 ) ≤ r ( xψ 2 ) ∧ d ( y ) ψ 2 , etc.) ≤ ( x | r ( x ) ∧ d ( y )) ψ 2 ⊗ ( r ( x ) ∧ d ( y ) | y ) ψ 2 , b y ICP4 ≤ (( x | r ( x ) ∧ d ( y )) · ( r ( x ) ∧ d ( y ) | y )) ψ 2 , b y ICP1 = ( x ⊗ y ) ψ 2 . (5.2) EXTENDING THE E SN THEOREM 15 But x ⊗ y = ( sψ 1 ) ⊗ ( tψ 1 ) ≤ ( s · t ) ψ 1 , b y (ICP1). Then, b y (ICP3) for ψ 2 , ( sψ 1 ψ 2 ) ⊗ ( tψ 1 ψ 2 ) ≤ ( s · t ) ψ 1 ψ 2 .  Th us inductiv e categories and inductiv e category prefunctors f orm a category . W e now prov e that the functions of Definitions 5.2 and 5.4 are indeed connected in the desired w a y: Prop osition 5.7. L et θ : S → T b e an or der e d ( ∧ , r ) -pr em orphism of r estriction semigr oups S and T . We d e fine Θ := C ( θ ) : C ( S ) → C ( T ) to b e the sam e function on the underlying se ts. Then Θ is an inductive c ate gory pr efunc tor with r esp e ct to the r e stricte d p r o ducts in C ( S ) and C ( T ) . Pr o of. (ICP1) Supp ose that ∃ s · t . Then ( s Θ) ⊗ ( t Θ) = ( sθ )( tθ ) (the pro duct in T ) ≤ ( st ) θ = ( s · t ) θ , since ∃ s · t = ( s · t )Θ . (ICP2) W e hav e d ( s Θ) = ( sθ ) + ≤ s + θ = d ( s )Θ. Similar ly , r ( s Θ) ≤ r ( s )Θ. (ICP3) This follows from Lemma 4.6. (ICP4) Let a ∈ C ( S ) and f ∈ C ( S ) o = E . Then a Θ | r ( a Θ) ∧ f Θ = ( aθ )( aθ ) ∗ ( f θ ) = ( aθ )( f θ ) ≤ ( af ) θ = ( aa ∗ f ) θ = ( a | r ( a ) ∧ f )Θ . Similarly , e Θ ∧ d ( a Θ) | a Θ ≤ ( e ∧ d ( a ) | a )Θ, for e ∈ E .  Prop osition 5.8. L et ψ : C → D b e an inductive c ate gory pr efunctor. We define Ψ := S ( ψ ) : S ( C ) → S ( D ) to b e the sa m e function on the underlying sets. Then Ψ is an or der e d ( ∧ , r ) -pr emorphism with r esp e ct to the pseudopr o ducts in S ( C ) and S ( D ) . Pr o of. ( ∧ 1) L et s, t ∈ S ( C ). Then ( s Ψ) ⊗ ( t Ψ) = ( sψ ) ⊗ ( tψ ) ≤ ( s ⊗ t ) ψ = ( s ⊗ t )Ψ, b y an argument iden tical t o that found b etw een lines (5.1) and (5.2) in Lemma 5.6. ( ∧ 2) W e hav e: ( s Ψ ) + = d ( sψ ) ≤ d ( s ) ψ = s + Ψ. Similarly , ( s Ψ) ∗ ≤ s ∗ Ψ. Finally , it follows from Lemma 4.6 t ha t Ψ is order-preserving.  Once again, it is easy to see that if θ : S → T is an ordered ( ∧ , r )-premorphism of restriction semigroups and ψ : C → D is an inductiv e catego r y prefunctor, then S ( C ( θ ) ) = θ and C ( S ( ψ )) = ψ . F urthermore, if θ ′ : T → T ′ is ano t her ordered ( ∧ , r )-premorphism of restriction semigroups, and ψ ′ : D → D ′ is another inductiv e category prefunctor of inductiv e categories, t hen C ( θ θ ′ ) = C ( θ ) C ( θ ′ ) and S ( φφ ′ ) = S ( φ ) S ( φ ′ ). W e hav e therefore pro v ed Theorem 5.1. 5.2. Strong ( ∧ , r )-premorphisms. As w e noted at the b eginning o f the sec- tion, in order to make the connection with the inv erse case, we m ust now in- tro duce t he in termediate step of str ong ( ∧ , r )-premorphisms b etw een restriction semigroups. Our goa l f o r the remainder of this section is the pro of o f the fo llo wing corollary to Theorem 5.1: 16 CHRISTOPHER HO LLINGS Theorem 5.9. The c ate gory of r estriction semigr oups and str on g ( ∧ , r ) -pr e- morphisms is isomorph i c to the c a te gory of ind uctive c ate gories and str ong in- ductive c ate gory pr efunctors. Once again, it only remains to deal with the ‘arrows’ part. In the study of partial actions of one-sided restriction semigroups [1 0], it is t he notion of a ‘strong premorphism’ which has prov ed most useful. W e mak e the follo wing definition in the tw o-sided case: Definition 5.10. Let S and T b e restriction semigroups. A ( ∧ , r )-premorphism θ : S → T is called str on g if ( ∧ 1) ′ ( sθ )( tθ ) = ( sθ ) + ( st ) θ = ( st ) θ ( tθ ) ∗ . It is clear that condition ( ∧ 1) follo ws from condition ( ∧ 1) ′ , b y (2.2). F urther- more, w e deduce the following fr o m [10 , Lemma 2 .10(4)]: Lemma 5.11. A str ong ( ∧ , r ) -pr emorp h ism θ : S → T b etwe en r e s triction semi- gr oups is or der-pr eserving. W e therefore drop all explicit men tio n of order-preserv ation from here on. Lemma 5.12. The c omp osition of two str ong ( ∧ , r ) -pr emorphism s is a str on g ( ∧ , r ) -p r emorphism. Pr o of. Let θ 1 : U → V and θ 2 : V → W b e strong ( ∧ , r )- premorphisms of restriction semigroups U , V and W . Condition ( ∧ 2) is immediate: ( sθ 1 θ 2 ) + ≤ (( sθ 1 ) + ) θ 2 ≤ s + θ 1 θ 2 . Similarly , ( sθ 1 θ 2 ) ∗ ≤ s ∗ θ 1 θ 2 . F or ( ∧ 1) ′ , we hav e ( sθ 1 θ 2 )( tθ 1 θ 2 ) = ( sθ 1 θ 2 ) + (( sθ 1 )( tθ 1 )) θ 2 , using ( ∧ 1) ′ for θ 2 . No w, ( sθ 1 θ 2 ) + ≤ (( sθ 1 ) + ) θ 2 = ⇒ ( sθ 1 θ 2 ) + ≤ (( sθ 1 ) + ) θ + 2 = ⇒ ( sθ 1 θ 2 ) + = ( sθ 1 θ 2 ) + (( sθ 1 ) + ) θ + 2 , b y (2.3), so ( sθ 1 θ 2 )( tθ 1 θ 2 ) = ( sθ 1 θ 2 ) + (( sθ 1 ) + ) θ + 2 (( sθ 1 )( tθ 1 )) θ 2 = ( sθ 1 θ 2 ) + (( sθ 1 ) + ) θ + 2 (( sθ 1 ) + ( st ) θ 1 ) θ 2 (using ( ∧ 1) ′ for θ 1 ) = ( sθ 1 θ 2 ) + (( sθ 1 ) + ) θ 2 ( st ) θ 1 θ 2 (applying ( aθ 2 )( bθ 2 ) = ( aθ 2 ) + ( ab ) θ 2 with a = ( sθ 1 ) + and b = ( st ) θ 1 ) = ( sθ 1 θ 2 ) + ( st ) θ 1 θ 2 , since ( sθ 1 θ 2 ) + ≤ (( sθ 1 ) + ) θ 2 . Similarly , ( sθ 1 θ 2 )( tθ 1 θ 2 ) = ( st ) θ 1 θ 2 ( sθ 1 θ 2 ) ∗ .  EXTENDING THE E SN THEOREM 17 Th us restriction semigroups together with strong ( ∧ , r )-premorphisms form a category . W e no w define a function b et w een inductiv e categories which will corresp ond to a strong ( ∧ , r )-premorphism b etw een restriction semigroups. Definition 5.13. An inductiv e cat ego ry prefunctor ψ : C → D will b e called str ong if (ICP5) (a) d (( s ψ ) ⊗ ( tψ )) = d ( sψ ) ∧ d (( s ⊗ t ) ψ ); (b) r (( sψ ) ⊗ ( tψ )) = r (( s ⊗ t ) ψ ) ∧ r ( tψ ). W e note the follow ing: Lemma 5.14. Condition (ICP4) fol lows fr om (ICP5). Pr o of. Let ψ : C → D b e a function b etw een inductiv e categories whic h satisfies conditions (ICP1)–(ICP3), plus (ICP5). W e consider (ICP4) ′ (a) and o bserv e that, for a ∈ C and f ∈ C o , aψ ⊗ f ψ ≤ ( a ⊗ f ) ψ in D ⇐ ⇒ aψ ⊗ f ψ ≤ ( a ⊗ f ) ψ in S ( D ) ⇐ ⇒ aψ ⊗ f ψ = ( a ⊗ f ) ψ ⊗ r ( aψ ⊗ f ψ ) ⇐ ⇒ aψ ⊗ f ψ = ( a ⊗ f ) ψ ⊗ ( r ( aψ ) ∧ f ψ ) , (5.3) using Lemma 3.7. W e will use Lemma 3.10, in conjunction with (ICP5), to demonstrate the equality (5.3). W e not e first of all that aψ ⊗ f ψ ≤ aψ and also that ( a ⊗ f ) ψ ⊗ ( r ( aψ ) ∧ f ψ ) ≤ ( a ⊗ f ) ψ ≤ aψ , since a ⊗ f ≤ a and ψ is order- pr eserving. It remains to sho w that eac h side of (5.3) has the same range. On the one hand, w e ha v e r ( aψ ⊗ f ψ ) = r (( a ⊗ f ) ψ ) ∧ r ( f ψ ) = r (( a ⊗ f ) ψ ) ∧ f ψ , b y (ICP5)(b). On t he other, using Lemma 3.7, ( a ⊗ f ) ψ ⊗ ( r ( aψ ) ∧ f ψ ) = ( a ⊗ f ) ψ | r (( a ⊗ f ) ψ ) ∧ r ( aψ ) ∧ f ψ , = ( a ⊗ f ) ψ | r (( a ⊗ f ) ψ ) ∧ f ψ , since r (( a ⊗ f ) ψ ) ≤ r ( aψ ). Th us r (( a ⊗ f ) ψ ⊗ ( r ( aψ ) ∧ f ψ )) = r (( a ⊗ f ) ψ ) ∧ f ψ = r ( aψ ⊗ f ψ ) , as required. W e conclude that (ICP4) ′ (a) holds. Part ( b) follo ws similarly .  Th us a strong inductive category prefunctor may b e regarded as a function defined b y conditions (ICP1)–(ICP3) and (ICP5) only . W e defer the pro of that the comp osition of t w o strong inductiv e cat ego ry pre- functors is a strong inductiv e category prefunctor un til after the fo llowing prop o- sitions, whic h pro v e that the functions of Definitions 5.10 a nd 5.13 are once again connected in the desired wa y: 18 CHRISTOPHER HO LLINGS Prop osition 5.15. L et θ : S → T b e a str ong ( ∧ , r ) -pr emorphism of r es triction semigr oups. We define Θ := C ( θ ) : C ( S ) → C ( T ) to b e the same function on the underlying sets. Then Θ is a str ong inductive c ate gory pr e f unc tor with r esp e ct to the r estricte d pr o ducts in C ( S ) and C ( T ) . Pr o of. By Prop osition 5.7, Θ is an inductiv e category prefunctor. (ICP5)(a) W e hav e d (( s Θ) ⊗ ( t Θ)) = (( sθ )( tθ )) + = (( sθ ) + ( st ) θ ) + = (( sθ ) + ( st ) θ + ) + = ( sθ ) + ( st ) θ + = d ( s Θ) ∧ d (( s ⊗ t )Θ) . P art (b) is similar.  Prop osition 5.16. L et ψ : C → D b e a str ong inductive c ate gory pr ef unc tor. We d e fine Ψ := S ( ψ ) : S ( C ) → S ( D ) to b e the same function on the underlying sets. Then Ψ is a str o n g ( ∧ , r ) -pr emorphism with r es p e ct to the pseudop r o ducts in S ( C ) and S ( D ) . Pr o of. By Prop osition 5.8, Ψ is an ordered ( ∧ , r )- premorphism. W e kno w that ( s Ψ) ⊗ ( t Ψ) ≤ ( s ⊗ t )Ψ and that ( s Ψ) + ⊗ ( s ⊗ t )Ψ ≤ ( s ⊗ t )Ψ, b y (2.2). W e will use Lemma 3.10 to sho w that condition ( ∧ 1) ′ holds. It remains to sho w that b oth sides of the desired equalit y ha v e the same domain. F r o m (ICP5), w e ha v e: d (( s Ψ) ⊗ ( t Ψ)) = d ( s Ψ) ∧ d (( s ⊗ t )Ψ) = d ( d ( s Ψ) ∧ d (( s ⊗ t )Ψ)) = (( s Ψ) + ⊗ ( s ⊗ t )Ψ + ) + = (( s Ψ) + ⊗ ( s ⊗ t )Ψ) + = d (( s Ψ) + ⊗ ( s ⊗ t )Ψ) . So b y Lemma 3.10, ( s Ψ) ⊗ ( t Ψ) = ( s Ψ) + ⊗ ( s ⊗ t )Ψ. By a similar argumen t, ( s Ψ) ⊗ ( t Ψ) = ( s ⊗ t )Ψ ⊗ ( t Ψ) ∗ .  It is clear that if θ : S → T is a strong ( ∧ , r )-premorphism a nd ψ : C → T is a strong inductiv e category prefunctor, then S ( C ( θ )) = θ and C ( S ( ψ )) = ψ . F urthermore, if θ ′ : T → T ′ is another strong ( ∧ , r )- premorphism, then C ( θ θ ′ ) = C ( θ ) C ( θ ′ ). In order to complete the pro o f of Theorem 5.9 , w e m ust pro v e that the comp osition of t w o strong inductiv e category prefunctors is also a strong inductiv e category prefunctor, thereby sho wing that inductiv e cat ego ries together with strong inductive category prefunctors do indeed form a catego ry . Lemma 5.17. The c omp osition of two str o ng i n ductive c ate gory p r efunctors is a str ong inductive c ate go ry pr efunctor. Pr o of. W e tak e an indirect a pproac h using the preceding prop osition. Let ψ 1 : U → V and ψ 2 : V → W be strong inductive catego r y prefunctors b etw een in- ductiv e categories U , V and W . By Prop osition 5.16, w e can construct strong ( ∧ , r )- premorphisms S ( ψ 1 ) : S ( U ) → S ( V ) a nd S ( ψ 2 ) : S ( V ) → S ( W ). Then S ( ψ 1 ) S ( ψ 2 ) : S ( U ) → S ( W ) is a strong ( ∧ , r )-premorphism, b y Lemma 5.1 2. EXTENDING THE E SN THEOREM 19 No w, ψ 1 ψ 2 is certainly an inductiv e category prefunctor ( but is not necessarily strong), in whic h case, S ( ψ 1 ψ 2 ) is an ordered ( ∧ , r )-premorphism, b y Prop o si- tion 5 .8. Let ‘ ∼ ’ denote the relationship ‘...is t he same function on the under- lying sets as...’. Then ψ 1 ∼ S ( ψ 1 ) a nd ψ 2 ∼ S ( ψ 2 ), so ψ 1 ψ 2 ∼ S ( ψ 1 ) S ( ψ 2 ). But ψ 1 ψ 2 ∼ S ( ψ 1 ψ 2 ). W e deduce t hat S ( ψ 1 ) S ( ψ 2 ) = S ( ψ 1 ψ 2 ), hence S ( ψ 1 ψ 2 ) is a strong ( ∧ , r )-premorphism. Then C ( S ( ψ 1 ψ 2 )) = ψ 1 ψ 2 is a strong inductiv e category prefunctor, by Prop osition 5.15.  Th us Theorem 5.9 is prov ed. 6. The inverse case W e will now deduce a corollary to Theorem 5.9 in the inv erse case. Our notion of ‘ ∧ -premorphism’ will simply b e t ha t of Definition 1.2. In addition, if suc h a ( ∧ , i )-premorphism is order-preserving, then we will call it an or d e r e d ( ∧ , i )- premorphism. W e will prov e the fo llo wing: Theorem 6.1. T he c ate gory of inverse semi g r oups and or der e d ( ∧ , i ) -pr emorph- isms is isomorphic to the c ate go ry of in ductive gr oup oids and or der e d gr oup oid pr emorphism s. Note that the ‘ob jects’ part o f the ab o ve theorem is tak en care of b y the commen ts at the end o f Section 4. W e deduce the following fro m [10, Lemma 2 .12]: Lemma 6.2. L et θ : S → T b e a m apping b etwe en inverse semigr oups. Then θ is an or der e d ( ∧ , i ) -pr emorphis m if, and only if, it is a str ong ( ∧ , r ) -pr emorphism of r estriction semigr oups. W e now , of course, need a corresponding function b etw een inductiv e group oids. In his study o f the partial actions of inductiv e g r o up oids, G ilb ert [7] has emplo y ed the follow ing definition: Definition 6.3. [7, p. 184] A function ψ : G → H b et wee n inductiv e group oids will b e called an or der e d gr oup oid pr emo rphism if the f o llo wing conditions are satisfied: (ICP1) if the pro duct g · h is defined in G , then ( g ψ ) ⊗ ( hψ ) ≤ ( g · h ) ψ ; (IGP) ( g ψ ) − 1 = g − 1 ψ ; (ICP3) if g ≤ h in G , then g ψ ≤ hψ in H . W e wan t to sho w tha t Gilb ert’s ordered group oid premorphisms are a sp ecial case of our strong inductive category prefunctors. In particular, a s a first step to w a rds establishing an analog ue of Lemma 6 .2, w e need to sho w that an ordered group oid premorphism satisfies condition (ICP5). Unfortunately , an elemen tar y pro of of this do es not seem to b e forthcoming; instead, w e provide a pro of whic h emplo ys the mac hinery of the Szendrei expansion of an inductiv e group oid, as in tro duced in Section 3 .2 , to gether with the following result of G ilb ert: 20 CHRISTOPHER HO LLINGS Theorem 6.4. [7, Theorem 4.4 & Prop osition 4.6] L et G and H b e inductive gr oup oids. If ψ : G → H is an or der e d gr oup o i d pr emorphism, then ther e ex i s ts a unique inductive functor ψ : Sz ( G ) → H such that ιψ = ψ . Conversely, if ψ : Sz( G ) → H is an inductive functor, then ψ := ιψ is an or der e d gr o up oid pr emorphism . Lemma 6.5. L et ψ : G → H b e an or der e d gr oup oid pr emorphism. Then ψ satisfies c ondition (ICP5). Pr o of. By Theorem 6.4, ψ may b e decomp osed as ψ = ι ψ , where ι : G → Sz( G ) is as in Section 3.2 and ψ : Sz ( G ) → H is an inductiv e functor. Note that an inductiv e functor preserv es pro ducts, doma ins, ranges, ordering, restrictions, corestrictions and meets. W e will demonstrate t hat (ICP5)(a) holds; part (b) is similar. Let s, t ∈ G . Then sψ = sι ψ = ( { d ( s ) , s } , s ) ψ and tψ = tιψ = ( { d ( t ) , t } , t ) ψ . W e ha v e sψ ⊗ tψ = ( { d ( s ) , s } , s ) ψ ⊗ ( { d ( t ) , t } , t ) ψ = [ ( { d ( s ) , s } , s ) ⊗ ( { d ( t ) , t } , t ) ] ψ = ( ∆ | { d ( s ) , s } ∪ s ⊗ { d ( t ) , t } , s ⊗ t ) ψ , b y (3.3), where ∆ = d ( s | r ( s ) ∧ d ( t )) = d ( s ⊗ t ). No w, ∆ ≤ d ( s ), so ∆ | d ( s ) = ∆. Also, s ⊗ d ( t ) = s | r ( s ) ∧ d ( t ) = ∆ | s , by Lemma 3 .1 0, so sψ ⊗ tψ = ( { ∆ , ∆ | s, s ⊗ t } , s ⊗ t ) ψ . Then, b y (3.1), d ( sψ ⊗ tψ ) = ( { ∆ , ∆ | s, s ⊗ t } , ∆ ) ψ . On the other hand, ( s ⊗ t ) ψ = ( s ⊗ t ) ι ψ = ( { d ( s ⊗ t ) , s ⊗ t } , s ⊗ t ) ψ = ( { ∆ , s ⊗ t } , s ⊗ t ) ψ , so that d (( s ⊗ t ) ψ ) = ( { ∆ , s ⊗ t } , ∆ ) ψ . Note that d ( sψ ) = ( { d ( s ) , s } , d ( s )) ψ . W e ha v e d ( sψ ) ∧ d (( s ⊗ t ) ψ ) = ( { d ( s ) , s } , d ( s ) ) ψ ∧ ( { ∆ , s ⊗ t } , ∆ ) ψ = [ ( { d ( s ) , s } , s ) , d ( s ) ) ∧ ( { ∆ , s ⊗ t } , ∆ ) ] ψ = ( ∆ | { d ( s ) , s, ∆ , s ⊗ t } , ∆ ) ψ ( by (3.2)) = ( { ∆ , ∆ | s, s ⊗ t } , ∆ ) ψ (since ∆ | ( s ⊗ t ) = s ⊗ t , by Lemma 3 .4 ) = d ( sψ ⊗ tψ ) , as required.  W e note the follow ing: EXTENDING THE E SN THEOREM 21 Lemma 6.6. [7 , Lemma 4.2] L et ψ : G → H b e an or der e d gr oup oid pr em o rphism. Then, for any g ∈ G , d ( g ψ ) ≤ d ( g ) ψ and r ( g ψ ) ≤ r ( g ) ψ . That is, an o rdered g roup oid premorphism satisfies condition (ICP2). Th us, b y the tw o preceding lemmas, every or dered group o id premorphism is a strong inductiv e category prefunctor. Lemma 6.7. The c om p osition of two or der e d gr oup o id pr emorphisms is an o r- der e d gr oup oid pr emorphism. Pr o of. The comp osition of tw o ordered group o id premorphisms is certainly a strong inductiv e category prefunctor, b y Lemma 5.17. Condition (IG P) follows easily .  Inductiv e group oids together with o r dered group oid premorphisms therefore form a category . Prop osition 6.8. L et θ : S → T b e an or der e d ( ∧ , i ) -pr e m orphism. We define Θ := C ( θ ) : C ( S ) → C ( T ) to b e the same function on the underlying s e ts. Then Θ is an or d er e d gr o up oid pr emorphi sm with r esp e ct to the r estricte d pr o d ucts in C ( S ) and C ( T ) . Pr o of. By Lemma 6.2, θ is a strong ( ∧ , r )-premorphism. Then Θ is a strong inductiv e category prefunctor, b y Prop osition 5.15. It remains to sho w tha t Θ satisfies (IGP): ( g Θ) − 1 = ( g θ ) − 1 = g − 1 θ = g − 1 Θ .  Prop osition 6.9. L et ψ : G → H b e an or der e d gr oup oid pr emorp h ism. We define Ψ := S ( ψ ) : S ( G ) → S ( H ) to b e the same function on the underlying s e ts. Then Ψ is an or der e d ( ∧ , i ) -pr emorp h ism with r esp e ct to the pseudopr o ducts in S ( G ) and S ( H ) . Pr o of. W e kno w that ψ is a strong inductiv e category prefunctor. Then, by Prop osition 5.16, Ψ is a strong ( ∧ , r ) -premorphism. It remains to sho w tha t condition ( ∧ 2) ′ is satisfied: ( g Ψ) − 1 = ( g ψ ) − 1 = g − 1 ψ = g − 1 Ψ , a s required.  Th us ordered group oid premorphisms and ordered ( ∧ , i )-premorphisms ar e con- nected in the manner required to prov e Theorem 6.1. W e can no w complete our inductiv e gro up oid analo gue of Lemma 6.2: Lemma 6.10. L et ψ : G → H b e a func tion b etwe en inductive gr oup oids. The n ψ is a n or der e d gr o up oid pr emorphism if, and only if, it is a str ong in d uctive c ate gory pr efunctor. Pr o of. ( ⇒ ) Supp ose that ψ : G → H is an or dered gro up oid premorphism. It follo ws from Lemmas 6 .5 and 6.6 that ψ is a strong inductiv e category prefunctor. ( ⇐ ) Supp ose that ψ : G → H is a strong inductiv e category prefunctor of inductiv e group oids. By Prop o sition 5.16, we hav e a strong ( ∧ , r )- premorphism S ( ψ ) : S ( G ) → S ( H ). But this is a strong ( ∧ , r )- premorphism o f in v erse semi- groups, so, by Lemma 6.2, it is an ordered ( ∧ , i )-premorphism. Then C ( S ( ψ )) = ψ is a n ordered g r oup oid premorphism, b y Prop osition 6.8.  22 CHRISTOPHER HO LLINGS As ev er, it is clear that if θ : S → T is an ordered ( ∧ , i )-premorphism and ψ : G → H is an ordered group oid premorphism, then S ( C ( θ )) = θ and C ( S ( ψ )) = ψ . Also, if θ ′ : T → T ′ is anot her ordered ( ∧ , i )-premorphism of in v erse semigroups, and ψ ′ : H → H ′ is another ordered group o id premorphism, then C ( θ θ ′ ) = C ( θ ) C ( θ ′ ) and S ( ψ ψ ′ ) = S ( ψ ) S ( ψ ′ ). W e ha v e prov ed Theorem 6.1. Let RE ST denote a category whose o b jects are r estriction semigroups; sub- scripts of ‘mor’, ‘strong ’, ‘ ∧ ’ and ‘ ∨ ’ will denote t ha t the a rro ws of the category are (2,1,1)- morphisms, strong ( ∧ , r )-premorphisms, ordered ( ∧ , r )-premorphisms and ( ∨ , r ) - premorphisms, resp ectiv ely . Th us, for example, REST strong denotes the category of restriction semigroups and strong ( ∧ , r )-premorphisms. Simi- larly , INV will denote a category whose ob jects are in verse semigroups; the subscripts applied here will b e ‘mor’, ‘ ∧ ’ and ‘ ∨ ’, denoting mor phisms, ordered ( ∧ , i )-premorphisms and ( ∨ , i )-premorphisms, respectiv ely . A category whose ob- jects are inductiv e categories or inductiv e group oids will b e denoted by IC or IG , as appro priate. A subscript of ‘ord’ or ‘ind’ will denote o rdered functors or induc- tiv e functors, resp ectiv ely , whilst ‘pre’, ‘strong’ a nd ‘ogp’ will denote inductive category prefunctors, strong inductiv e category prefunctors and ordered group oid premorphisms, resp ectiv ely . W e can no w summarise the connections b et w een the v arious categories in this pa p er in the fo llowing pa ir of Hasse diagrams: REST ∨ REST ∧ REST strong REST mor INV ∨ INV ∧ INV mor IC ord IC pre IC strong IC ind IG ord IG ogp IG ind Eac h categor y in the left-hand diagram is isomorphic to the corresp onding category in the right-hand diagra m, and vice v ersa. Reference s [1] S. Armstr ong, The structure of type A semig roups, S emigr oup F orum , 29 (198 4), 31 9 -336. [2] J.- C. Birg et and J. Rho des, Almost finite expansions o f a rbitrary semigroups, Journal of Pur e and Applie d Algebr a , 32 (1984), 239 -287. [3] J. R. B. Co ck ett and E. Manes, Bo olean and clas sical restriction ca teg ories, preprint, 2007 . [4] J. F ount ain, A class of right PP monoids, Quarterly Journal of Mathematics, Oxfor d (2) , 28 (1977), 285-300 . [5] J. F ountain, Adequate semig r oups, Pr o c e e dings of the Ed inbur gh Mathematic al So ciety (2) , 22 (1979), 113-125 . EXTENDING THE E SN THEOREM 23 [6] J. F ountain a nd G. M. S. Gomes, The Szendrei expansion o f a semigro up, Mathematika , 37 (1990), 251-260 . [7] N. D. Gilbe rt, Actions and expansions of o rdered group oids, Journal of Pure and Applie d Alge br a , 198 (200 5), 175-195. [8] G. M. S. Gomes and V. Gould, Finite pr op er cov ers in a class of finite s e mig roups w ith commuting idemp otents, Semigr oup F orum , 66 (2003), 433-45 4 . [9] V. Gould, (W ea kly) left E -ample semigro ups , http:/ /www- users.york.ac.uk/ ∼ varg1/finitela.ps [10] V. Gould and C. Hollings, P artial actions of in verse and w eakly left E -ample semig roups, to appea r in Journal of the Austr alian Mathematic al So ciety . [11] C. Hollings, Partial Ac tions of Semigr oups and Monoid s , Ph.D. thesis, Univ ersit y of Y ork, 2007. [12] C. Hollings, P artial actions of monoids, Semigr oup F orum , 75 (2) (2007 ), 29 3-316 . [13] M. Jackson and T. Stokes, An invitation to C -semigroups, Semigr oup F orum , 62 (2 001), 279-3 10. [14] N. Jacobso n, Basic Algebr a , V o lume I I, W. H. F reema n and Co., San F r ancisco, 1980. [15] M. V. La wson, Semigroups and ordered ca tegories I: the r educed cas e , Journal of Algebr a , 141 (1991), 422-46 2. [16] M. V. Lawson, Inverse S emigr oups: The The ory of Partial Symmetries , W orld Scientific, 1998. [17] D. B. McAlister a nd N. R. Reilly , E -unitary cov ers for inverse semigroups, Pacific Journal of Mathematics , 6 8 (1977), 161-174 . [18] M. B. Szendr ei, A note on Birget-Rho des e xpansion of groups, Journal of Pur e and Applie d Alge br a , 58 (198 9 ), 9 3-99. Centr o de ´ Algebra da Universidade de Lisboa, A v. Prof. G ama Pinto 2, 1649- 003 Lisboa, Por tugal E-mail addr ess : cdh5 00@cii .fc.ul .pt