Trace- and pseudo-products: restriction-like semigroups with a band of projections
aa r X i v : . [ m a t h . R A ] F e b TRACE- AND PSEUDO-PRODUCTS: RESTRICTION-LIKESEMIGROUPS WITH A BAND OF PROJECTIONS
D. G. FITZGERALD & M. K. KINYON
Abstract.
We ascertain conditions and structures on categories and semigroups whichadmit the construction of pseudo-products and trace products respectively, makingtheir connection as precise as possible. This topic is modelled on the ESN Theoremand its generalization to ample semigroups. Unlike some other variants of ESN, itis self-dual (two-sided), and the condition of commuting projections is relaxed. Thecondition that projections form a band (are closed under multiplication) is shown to bea very natural one. One-sided reducts are considered, and compared to (generalized)D-semigroups. Finally the special case when the category is a groupoid is examined. Introduction
The distant aim of this paper is to enrich the study of a groupoid over a skew lattice ofidentities, begun by the first author in [4]. But since a skew lattice possesses two opera-tions, each of which confers a band structure, a necessary step is to study a groupoid, ormore generally a category, over a band of objects. That is the more immediate aim here,and locates the paper squarely in the tradition of the Ehresmann-Schein-Nambooripad(ESN) Theorem, which (in its various forms) asserts equivalence of certain kinds of cate-gories having extra structure with certain special kinds of semigroups. This equivalencecan help elucidate one kind of structure in terms of the other; for example, the ESNTheorem has recently been used to design a relatively efficient enumeration of finiteinverse semigroups [13]. We must take a little further space to summarise the locationof the present paper in this extensive field.Ehresmann’s use of pseudogroups in describing symmetries in differential geometry,and Wagner’s and Schein’s similar use of groupoids and inverse semigroups, are describedby Lawson in Sections 1.1–3 of the book [11], with a history of the ideas behind theirrespective contributions outlined in Section 4.4 of the same. The interested reader isreferred to those sections, to the extended account of Hollings [8], or to the originalworks cited therein. Suffice it to say here that the constructions of the trace product ina suitable semigroup, and the pseudoproduct in a suitable groupoid, are central to thesuccess of the Theorem.Nambooripad’s concern was with regular semigroups, and the constructions in hisworks are more general and more complex; see the survey article of Szendrei [20], es-pecially Section 4. His point of departure was the description by Miller and Cliffordof a completely 0-simple (regular) semigroup as a Rees groupoid [14]; he described andformulated the notion of a bi-ordered set (characterising the set of idempotents, andinitiating a whole field of research which continues today). This he used in conjunction
Date : February 22, 2021. with a particular kind of ordered groupoid generalizing Rees groupoids (the inductivegroupoids in his terminology—a different use of the term than elsewhere), to formulatehis extension of the Theorem, incidentally making it fully categorical in flavour. Law-son [12] later extended the use of ordered groupoids to ordered (small) categories, andthis kind of generalization is also followed in the paper to hand.All these authors were concerned to explore the conditions which made the correspon-dence work; of course there are many “answers” possible. This paper has the declaredaim of tackling this question in the context of the idempotents or projections forminga band (it will be seen that this condition arises naturally). Thus the context here isbroader than the semilattice of idempotents of Ehresmann and Schein, but not as broadas Nambooripad’s bi-ordered sets. The semigroups we consider are a particular classof the P -semiabundant semigroups of Section 1 in Lawson [12], but distinct from theEhresmann semigroups of Section 3, which are the main focus of that paper.It was realised, particularly in work of the ‘York school’, that the field overlaps (andhas been extended to) other semigroup properties, some of which arose in the theory ofsemigroup acts; notably, it was refreshed and organised by Lawson. Its ramifications andnotation have been rationalized in the unpublished but very influential Notes of Gould [5];Hollings [7] surveyed its reach and history. Again the reader seeking more detail isreferred to these surveys. Most recently, contributions relevant to the present paperhave been made by Jones [9], Kudryavtseva [10] and El-Qallali [3]. Jones’s [9] is close inspirit to ours on the semigroup side, though we also consider equivalent categories. Jonesrelativises the Ehresmann-style axioms to a specified set P of commuting projections;in a generalization, he includes in his scheme the regular ∗ -semigroups of Nordahl andScheiblich [15], in which P is generally not closed under multiplication. In contrast toboth, our P is always a band.An important distinction among all these studies is between one-sided and two-sidedconditions. Gould and Stokes [6, 17, 18] have addressed corresponding questions for theone-sided case, in a strand which goes back to Cockett and Lack’s restriction categories [2]. Because of our concern with categories and the ESN theorem, our approach is mainlytwo-sided, though we are able also to make a small contribution in the one-sided theoryin Sections 4 and 6 below. All that said, in the end we aim to understand the structuresthat arise when a band or a skew lattice is extended in a manner analogous with inversesemigroup extensions of a semilattice—or a lattice, as in boolean inverse semigroups.2. Small categories
The system ( C, ◦ , + , − ), where C is a set, ◦ a partially defined binary operation C × C → C , and + , − are unary maps C → C , is a (small) category if it satisfies theaxioms x + ◦ x = x = x ◦ x − ;(2.1a) ( x + ) − = x + , ( x − ) + = x − ;(2.1b) x ◦ y is defined precisely when x − = y + ;(2.1c) when x − = y + , ( x ◦ y ) + = x + and ( x ◦ y ) − = y − ;(2.1d) ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) whenever the products are defined . (2.1e) RACE- AND PSEUDO-PRODUCTS 3
The set C + = { e ∈ C : e = e + } = C − is the set of identities or objects . Axiom 2.1b)implies ( x + ) + = (( x + ) − ) + = ( x + ) − = x + and similarly or dually, ( x − ) − = x − . For e ∈ C + and x ∈ C , e ◦ x is defined if and only if x + = e − = e and then e ◦ x = x ; and adual statement holds for x ◦ e .3. Transcription categories A transcription category is a category as above, endowed with two more maps C × C + → C , called transcription maps and written as follows: for e, f ∈ C + and x ∈ C , e | x, x | f denote arrows (members of C ) satisfying the axioms below, which are based onthose for ordered or inductive categories. e | f = e | f ;(3.1a) x + | x = x = x | x − ;(3.1b) e | ( f | x ) = ( e | f ) | x and its dual x | ( e | f ) = ( x | e ) | f ;(3.1c) e | ( x ◦ y ) = ( e | x ) ◦ ( ( e | x ) − | y ) and its dual ( x ◦ y ) | f = ( x | ( y | f ) + ) ◦ y | f ;(3.1d) ( e | x ) + = e | x + and its dual ( x | f ) − = x − | f ;(3.1e) ( e | x ) | f = e | ( x | f ) . (3.1f)Axiom (3.1a) ensures the two maps agree on C + × C + and by (3.1e) map C + × C + → C + .Axiom (3.1b) implies e | e = e and so with associativity from (3.1c) we have that C + isa band under the operation | · : ( e, f ) e | f . This band acts on the set C on both leftand right by (3.1c), interacting with the category operations as per (3.1d, e), and thetwo actions are linked by (3.1f).Observe that the natural-looking equation ( e | x ) − = ( e | x ) − | x − = ( e | x ) − | x − and itsdual ( x | f ) + = x + | ( x | f ) + = x + | ( x | f ) + are consequences of certain of these axioms: put y = x − in (3.1d), so the compositions are defined and e | x = e | ( x ◦ x − )=( e | x ) ◦ ( ( e | x ) − | x − ),whence(3.2) ( e | x ) − . d = ( ( e | x ) − | x − ) − . a = ( e | x ) − | x − . Given a transcription category ( C, ◦ , + , − ) or C for short, we define a pseudoproduct ⊗ as follows. For x, y ∈ C , define(3.3) x ⊗ y = ( x | y + ) ◦ ( x − | y ) , the right hand side being defined in C since( x | y + ) − = x − | y + = x − | y + = ( x − | y ) + by (3.1e) and (3.1a). Note that ⊗ extends both ◦ and the band operation | · when theyare defined: Lemma 3.1. i) If x ◦ y is defined then x ⊗ y = x ◦ y ;ii) if e, f ∈ C + , e ⊗ f = ( e | f ) ◦ ( e | f ) = e | f ;iii) ( x ⊗ y ) + = ( x | y + ) + and ( x ⊗ y ) − = ( x − | y ) − .Proof. (i) If x − = y + , then x | y + = x | x − = x , by (3.1b), and dually x − | y = y ; the resultfollows from the definition in equation (3.3). D. G. FITZGERALD & M. K. KINYON (ii) f ◦ f = f and by (3.1d) we have e | f = ( e | f ) ◦ ( ( e | f ) − | f ) = ( e | f ) ◦ ( ( e | f ) | f ) (using(3.1e)) and then with (3.1b, c) this equals ( e | f ) ◦ ( e | f ). Finally, e | f ∈ C − by (3.1e) so( e | f ) ◦ ( e | f ) = e | f .(iii) This comes from the definition and (2.1d). (cid:3) Theorem 3.2. If C is a transcription category then ( C, ⊗ ) is a semigroup.Proof. We have to show associativity of ⊗ . First, for any x, y, z ∈ C ,( x ◦ y ) ⊗ z .
3= ( x ◦ y ) | z + ◦ ( ( x ◦ y ) − | z ) 3 . d = ( x | ( y | z + ) + ◦ y | z + ) ◦ ( ( x ◦ y ) − | z )2 . e = x | ( y | z + ) + ◦ ( y | z + ◦ ( x ◦ y ) − | z ) 2 . d = x | ( y | z + ) + ◦ ( y | z + ◦ y − | z )3 . x | ( y | z + ) + ◦ ( y ⊗ z ) 3 . x | ( y ⊗ z ) + ◦ ( y ⊗ z )3 . x ⊗ ( y ⊗ z ) . Using this, we have ( x ⊗ y ) ⊗ z .
3= ( x | y + ◦ x − | y ) ⊗ z above = x | y + ⊗ ( y ⊗ z ) = x ⊗ ( y ⊗ z ),since ( x | y + ) | ( y ⊗ z ) + = ( x | y + ) | y + = x | ( y ⊗ z ) + . (cid:3) The semigroup ( C, ⊗ ) is denoted S ( C ) and has extra structure, which we proceed toexplore. 4. localizable semigroups A unary semigroup is an algebra ( S, · , +) of signature (2 ,
1) such that ( S, · ) is a semi-group, that is, x · ( y · z ) = ( x · y ) · z , together with a unary map + : S → S . A leftlocalizable semigroup is a unary semigroup such that the unary map + : S → S satisfies x + · x = x, (4.1a) ( x · y ) + = ( x · y + ) + , and(4.1b) x + · y + = ( x + · y ) + . (4.1c)A right localizable semigroup is an algebra ( S, · , − ) such that ( S opp , · , − ) is a leftlocalizable semigroup; that is, ( S, · , − ) satisfies the identities x · x − = x, (4.1d) ( x · y ) − = ( x − · y ) − , and(4.1e) x − · y − = ( x · y − ) − . (4.1f)A localizable semigroup is an algebra ( S, · , + , − ) of signature (2 , ,
1) such that ( S, · , +)is a left localizable and ( S, · , − ) a right localizable semigroup, and the unary operationsare linked by the identities ( x + ) − = x + and ( x − ) + = x − . (4.1g)These axioms are based on those for restriction semigroups, as well as the needs ofthe Theorems to follow. A more detailed comparison with restriction semigroups andone-sided variants is given in a later Section. For future use, let us observe that theseaxioms occur in dual pairs; to reduce repetition, we will often leave dual statementsimplicit. RACE- AND PSEUDO-PRODUCTS 5
Let S + = { x + : x ∈ S } , and note that (4.1g) implies S − = { x − : x ∈ S } = S + andalso ( x + ) + = (( x + ) − ) + = ( x + ) − = x + . As in restriction semigroups, members of S + are called projections . The next lemma records some useful relationships. Lemma 4.1.
Let ( S, · , +) be a unary semigroup. Theni) (4.1a) and c) imply ( S + , · ) is a band;ii) (4.1a), b) and c) imply ( x + ) + = x + ; andiii) if S + is a band, (4.1b) implies (4.1c).Proof. i) (4.1c) shows that x + · y + ∈ S + ; it also implies x + · x + = ( x + · x ) + which= x + by (4.1a).ii) Now from (4.1b) we have y + = ( y + · y + ) + = ( y + ) + .iii) Set x + for x in (4.1b) and use ( x + · y + ) + = x + · y + etc. by the band condition. (cid:3) This lemma also indicates that it is a natural and interesting condition that S + bea band (and not necessarily a semilattice). It is of further interest to point out thefollowing relations between axioms which involve both unary operations in (4.1). Lemma 4.2.
Let ( S, · , + , − ) be a semigroup with two unary operations in which (4.1a),d) and g) hold. Theni) (4.1c) and f ) together are equivalent to (4.2) ( x · y + ) − = x − · y + = ( x − · y ) + ; andii) if S + is a band, (4.1b) and e) together imply (4.2).Proof. i) Set y + in place of y in (4.1f) and use (4.1g) to obtain the first equation of(4.2). For the second (and dual) equation of (4.2), put x − for x in (4.1c). In theconverse direction, put y − for y in the first equation of (4.2) to get ( x · ( y − ) + ) − = x − · ( y − ) + , which is (4.1f) by (4.1g); and put x + for x in the second equation of(4.2) to get the dual (4.1c).ii) By Lemma 4.1(ii) and its dual, (4.1c) and f) hold. By part i), (4.2) follows. (cid:3) Lemma 4.3.
Let ( S, · , + , − ) be a localizable semigroup. Then for all x, y ∈ S ,i) x · y = x · y + · x − · y andii) ( x + · y ) − · y − = ( x + · y ) − .Proof. i) Now (4.1a) and d), taken with Lemma 4.1(i), show that x · y = x · x − · y + · y = x · ( x − · y + ) · y = x · x − · ( y + · x − ) · y + · y = x · y + · x − · y. ii) Using (4.1g) and f) twice, we have( x + · y ) − · y − = (( x + ) − · y ) − · y − = ( x + ) − · y − · y − = x + · y − = ( x + · y ) − , as required. (cid:3) Theorem 4.4. If C is a transcription category then S ( C ) is a localizable semigroup. D. G. FITZGERALD & M. K. KINYON
Proof.
Theorem 3.2 proved associativity of ⊗ , and axiom 4.1g) is 2.1b). We must verifythe remainder of axioms 4.1), and it is enough to prove 4.1a), b) and c), since theremainder are their duals. For 4.1a) we see x + ⊗ x = ( x + | x + ) ◦ ( ( x + ) − | x ) = x + ◦ x = x ,using 3.1b) and 2.1a). For 4.1b), axiom 2.1d) gives( x ⊗ y ) + = (( x | y + ) ◦ ( x − | y )) + = ( x | y + ) + , while( x ⊗ y + ) + = (( x | y + ) ◦ ( x − | y + )) + = (( x | y + ) + also . For 4.1c), ( x + ⊗ y ) + = ( x + | y + ) + = ( x + | y + ) = x + ⊗ y + , using Lemma 3.1. (cid:3) Given a localizable semigroup ( S, · , + , − ), define a partial binary mapping or com-position ◦ : S × S → S , called the trace product , and transcription maps defined asfollows. Definition 4.5. (i) The trace product x ◦ y is defined exactly when x − = y + , and then x ◦ y = x · y. (ii) Transcription maps for x ∈ S, e, f ∈ S + are defined by e | x = e · x and x | f = x · f . Complementing Theorem 4.4 we have
Theorem 4.6. If S is a localizable semigroup, then C ( S ) = ( S, ◦ , + , − ) is a transcriptioncategory.Proof. We first have to verify axioms (2.1a–e). Associativity in S guarantees (2.1e). Byinspection, (2.1a, b) and c) are already (4.1a), d) and g) and Definition 4.5(i). Now use x − = y + in 4.1(b) to obtain 2.1(d) as ( x ◦ y ) + = ( x · y ) + = ( x · y + ) + = ( x · x − ) + = x + ,and dually.So C ( S ) is a category and next we must deal with axioms (3.1). Of these, (a, b, c)are immediate from Definition 4.5(ii) and associativity of S . For 3.1(d) consider theLHS: e | ( x ◦ y ) = e · ( x · y ) with x − = y + . Now check the composition on the RHS isvalid: ( ( e | x ) − | y ) + = (( e · x ) − · y ) + = ( e · x ) − · y + = ( e · x ) − · x − , using (4.2). By Lemma4.1(iii), this last equals ( e · x ) − . Thus the composite ( e | x ) ◦ ( ( e | x ) − | y ) is defined, andequals ( e · x ) · ( e · x ) − · y = e · x · y , so 3.1(d) is satisfied. Wrapping up, 3.1(e) is immediatefrom 4.1(c), and 3.1(f) requires only associativity of S . (cid:3) Thus C and S are equivalent; in fact, Theorem 4.7. C ( S ( C )) = C and S ( C ( S )) = S .Proof. The constructions use the same base set S and maps + , − . Moreover, ◦ is arestriction of · , and ⊗ and · may be identified: for x, y ∈ S , x · y = x · y + · x − · y by Lemma4.3(i). Here, ( x · y + ) − = ( x − · y ) + by (4.2), whence x · y = ( x · y + ) ◦ ( x − · y ) = ( x | y + ) ◦ ( x − | y )by Definition 4.5, which equals x ⊗ y by (3.3). (cid:3) RACE- AND PSEUDO-PRODUCTS 7
Morphisms.
Let ( S, · , + , − ) and ( S ′ , · , + , − ) (in abuse of notation by overloading theoperation symbols) be localizable semigroups. A map φ : S → S ′ is a morphism oflocalizable semigroups or a ± -morphism if it preserves the operations in the usual sense,i.e., it is a semigroup morphism such that ( xφ ) + = ( x + ) φ and ( xφ ) − = ( x − ) φ . Similarly,let ( C, ◦ , + , − ) and ( C ′ , ◦ , + , − ) be transcription categories. A map ψ : C → C ′ is afunctor (morphism) of transcription categories if it is a (small) functor in the usual sense(thus ( xψ ) + = ( x + ) ψ , etc.) and preserves the transcription maps, i.e., ( x | e ) ψ = ( xψ ) | eψ ,etc. Theorem 4.8.
A map φ : S → S ′ is a morphism of localizable semigroups if and only ifit is a functor of transcription categories C ( S ) → C ( S ′ ) .Proof. Let φ : S → S ′ be a morphism, so that ( xφ ) + = ( x + ) φ in C ( S ′ ). If x ◦ y isdefined in C ( S ′ ), then ( xφ ) − = x − φ = y + φ = ( yφ ) + , so that xφ ◦ yφ is defined and= xφ · yφ = ( x · y ) φ = ( x ◦ y ) φ . If e ∈ S + , ( x | e ) φ = ( x · e ) φ = xφ · eφ = ( xφ | eφ )and similarly for the right transcription maps. Conversely, if φ : C → C ′ is a functor,( xφ ) ± = ( x ± ) φ and( x · y ) φ = ( x ⊗ y ) φ = [( x | y + ) ◦ ( x − | y )] φ = ( x | y + ) φ ◦ ( x − | y ) φ = ( xφ | y + φ ) ◦ ( x − φ | yφ ) = xφ ⊗ yφ = xφ · yφ, whence φ is a morphism S → S ′ . (cid:3) Corresponding to the definition of morphisms, we make the following
Definition 4.9. A localizable congruence δ on the localizable semigroup ( S, · , + , − ) isa congruence of the semigroup ( S, · ) for which ( x, y ) ∈ δ implies ( x + , y + ) , ( x − , y − ) ∈ δ .For brevity we also refer to such a congruence as a ± -congruence . Examples and related classes
For the remainder of the paper, we shall (except where emphasis is needed) omit thesymbol · and instead denote multiplication in the semigroup(s) by juxtaposition.5.1. Restriction semigroups.
We previously remarked on the similarity with restric-tion semigroups. In effect, in comparison with restriction semigroups, localizable semi-groups dispense with both (CP) and ample identities.By definition (see Kudryavtseva [10] for a recent paper which inspired the presentwork), S is a restriction semigroup if it satisfies (in the present notation) 4.1(a, c, d, f,g), the ample identities(5.1) xy + = ( xy ) + x, x − y = y ( xy ) − , and has commuting projections (CP). Incidentally, (CP) may be replaced by the condi-tion that S + be a band: Lemma 5.1.
If both ample identities (5.1) hold in a semigroup S satisfying 4.1(a,d,g)and S + S + ⊆ S + , then (CP) holds.Proof. Let p, q ∈ S + . Then with x = p = p − and y = q = q + we have both pq =( pq ) + p = ( pq ) p and pq = q · ( pq ) as identities in S + . Thus pq = pqp = qp . (cid:3) D. G. FITZGERALD & M. K. KINYON
In this context it is well-known that 4.1(b) is a consequence of 5.1:( xy + ) + = . [( xy ) + x ] + = . c ( xy ) + x + = CP x + ( xy ) + = . c ( x + xy ) + = . a ( xy ) + . The dual statement also holds, so any restriction semigroup, and in particular any inversesemigroup, satisfies all equations (4.1) and so is a localizable semigroup. This leaves openthe question of examining localizable semigroups with (CP).5.2.
Monoids.
Since any monoid S with identity 1 may be regarded as a restrictionsemigroup, it also is a localizable semigroup. One simply defines s + = s − = 1 for all s ∈ S . There is just one projection, and S is then called a reduced restriction monoid.(There are localizable monoids with multiple projections: for example, let S be anylocalizable semigroup and adjoin a new identity 1 with 1 + = 1 − = 1.)Reduced restriction monoids occur as subsemigroups in any localizable S : if e ∈ P ,then consider M e := { x ∈ S : x + = x − = e } . If x, y ∈ M e , ( xy ) + = ( xy + ) + = x + = e by(4.1b) and dually, ( xy ) − = e . So xy ∈ M e and M e is a subsemigroup of S , a localizablesemigroup in its own right, and a reduced restriction monoid. It coincides with themonoid of endomorphisms of object e in C ( S ).5.3. Orthocryptogroups.
Any band S is a localizable semigroup when we take s + = s − = s . More generally, let S be an orthocryptogroup , which is to say it is a unionof groups, is orthodox (its idempotents form a band) and its Green’s relation H is acongruence. By Theorem II.8.5 (iii) of the monograph [16], s t = ( st ) holds for all s, t ∈ S , where, as usual, s denotes the identity of the subgroup of S which contains s .Then, on setting s + = s − = s , one immediately verifies that 4.1a–g) all hold, and S isa localizable semigroup.Notwithstanding all the above, we still need non-trivial and non-artificial concreteexamples of localizable semigroups.6. One-sided reducts and generalized Green’s relations
In this section, we compare one-sided versions of localizable semigroups, consideredas a class of unary semigroups, with other such classes. Consider a left localizablesemigroup ( S, · , +) as defined in (4.1a, b, c), and let a relation on S be defined thus: s e ≤ R t if pt = t implies ps = s for all p ∈ S + . It is readily seen that e ≤ R is reflexive and transitive, and contains the right divisionrelation ( s = tx for some x ). We record some results for further reference; note thecrucial use of (4.1c), ( ps ) + = ps + . Lemma 6.1.
The following are equivalent:i) s e ≤ R t ;ii) s + e ≤ R t + ;iii) s + ≤ R t + ;iv) s = t + s . RACE- AND PSEUDO-PRODUCTS 9
Proof.
Let p ∈ S + . If (i) holds, pt + = t + implies pt = t and in turn ps = s , and ps + = ( ps ) + = s + , so (ii) holds. Also, since t + t + = t + , (ii) implies t + s + = s + , whichis (iii). Next, t + s + = s + implies t + s = s , so (iii) implies (iv). Finally, if t + s = s and pt = t , then ps = pt + s = ( pt ) + s = t + s = s, and (i) holds. (cid:3) Symmetrizing e ≤ R gives the ‘swung’ Green’s relation e R = { ( s, t ) : ps = s ⇔ pt = t } . Corollary 6.2.
For all s, t, u ∈ S , s e ≤ R t implies us e ≤ R ut , and so e R is a left congruence.Proof. Suppose s e ≤ R t ; by part (iv) of the Lemma, t + s = s . Consider ( ut ) + us =( ut + ) + us = ( ut + ) + ut + s = ( ut + ) s = us ; this shows us e ≤ R ut , and the rest follows. (cid:3) Let E be a distinguished set of idempotents of a unary semigroup ( S, · , +), and recallthat S is called weakly left E -abundant (Gould) or left E -semiabundant (Stokes) if every e R -class contains an element of E . It follows that left localizable semigroups are weaklyleft E -abundant with E = { s + : s ∈ S } = S + .However, the converse is not true: Example.
Let S be the semigroup with zero 0 generated by idempotents e, f suchthat f e = 0, writing ef = a for convenience. The diligent reader will verify with a littlelight work that S = { e, f, a, } , that E = E ( S ) = { e, f, } , and that the assignment e + = e = a + , f + = f, + = 0 satisfies y + x = x if and only if y + x + = x + . Thus x e R x + for all x ∈ S , and so S is weakly left E -abundant. However the correspondingunary operation x x + does not make ( S, · , +) left localizable, for E is not a band( ef = a / ∈ E ).A more stringent condition requires a unique element of E in each e R -class; this givesthe generalized D-semigroups studied by Stokes [18]. This class neither contains nor iscontained by that of left localizable semigroups. Indeed, a generalized D-semigroup S in which S + is not a band cannot be left localizable. On the other hand, if S is a band,an e R = R -class with distinct elements fails the uniqueness requirement, so S is leftlocalizable (with s + = s ) but not a generalized D-semigroup.The intersection of these two classes, the localizable generalized D-semigroups, isprecisely the class of D-semigroups [17]. This in turn contains (strictly) the generalizedleft restriction semigroups treated by Gould in [5].Note, however, that the unary projection map in an arbitrary left localizable semi-group ( S, · , +) may be modified to produce a generalized D-semigroup, as follows. Let C be a cross-section of the equivalence e R , and for s ∈ S , define s ⊕ to satisfy s ⊕ ∈ C and s + e R s ⊕ . It follows that s ⊕ s ⊕ = s ⊕ and s ⊕ s + = s + , whence s ⊕ s = s ⊕ s + s = s + s = s .That is, ( S, · , ⊕ ) is a generalized D-semigroup.7. Other structural features
An order.
We may define an analogue of the natural partial order on a restrictionsemigroup:
Definition 7.1.
Write s ✂ t if s = s + t = ts − . Recall that the natural (Mitsch) partial order ≤ M on any semigroup S is defined by s ≤ M t if s = t or there exist x, y ∈ S such that s = xt = ty = xty . Lemma 7.2.
The relation ✂ is a partial order on S which is contained in the naturalpartial order ≤ M , and coincides with it when restricted to the band S + .Proof. From Definition 7.1 it is clear that s ✂ s for all s ∈ S . If s ✂ t ✂ u , then s = s + t = s + t + u = ( s + t ) + u (by (4.1)c) = s + u , and dually s = us − , whence s ✂ u . Thenif s ✂ t ✂ s we have s = ts − and t = t + s = t + ts − = ts − = s . It is immediate from thedefinitions that s ✂ t implies s ≤ M t . When s = s + = s − , we have s ✂ t if and only if s = st = ts , as claimed. (cid:3) The order ✂ is the identity on any reduced restriction monoid, and so differs in generalfrom the Mitsch order. Actions and transformation representations.
The transcription structures lead toactions of S on the projections. These give representations of S by transformations,which naturally have weaker properties than their counterparts in the restiction semi-group setting. Definition 7.3.
For p ∈ S + and s, t ∈ S , let us write p s for ( ps ) − ∈ S + and dually, t p for ( tp ) + . Lemma 7.4.
For p ∈ S + and s, t ∈ S , ( p s ) t = p st and s ( t p ) = st p .Proof. ( p s ) t = ( p s t ) − = (( ps ) − t ) − = (( ps ) t ) − from axiom (4.1e), and so ( p s ) t =( p ( st )) − = p st . The second equation is dual. (cid:3) Thus we have defined a (right) action S + × S → S + , ( p, s ) p s . Let us write sδ : p p s for p ∈ S + , and from this define the map δ : s sδ . By Lemma 7.4,( st ) δ = sδ ◦ tδ , so δ is a semigroup morphism and a representation of S (as a plainsemigroup) in the transformation monoid T S + .There is dually a left action ( s, p ) s p = ( sp ) + and corresponding (anti-) repre-sentation γ . Notationally it is most convenient to write γ and γs as left mappings, γ : s γs , where γs ( p ) = s p = ( sp ) + , so that γst ( p ) = st p = ( stp ) + = γs ◦ γt ( p ), and γ manifests as a representation S → T S + . Thus the combination of γ and δ results in arepresentation γ ⊗ δ : S → T S + × T S + , s ( γs, sδ ).Let us note that γs ± ( p ) = ( s ± p ) + = s ± p , and dually ( p ) s ± δ = ps ± . (Here andlater we use ± with the meaning that it may be replaced in a statement by either +throughout or by − throughout.) Thus γs ± = λ s ± , the inner left translation on S + , and s ± δ = ρ s ± , the inner right translation on S + . As usual, we denote the band of inner left[right] translations of S + by Λ [P ], and the band of linked pairs of inner translations(bitranslations) ( λ p , ρ p ) of S + by Ω . Recall that S + ∼ = Ω ≤ Λ × P via the map p ( λ p , ρ p ), so that we may identify S + with Ω .The point of these remarks is that the definition of projections on the semigroup S ( γ ⊗ δ ) by means of ( γs, sδ ) ± = ( γs ± , s ± δ ) is well-defined if and only if ( γs, sδ ) = ( γt, tδ )implies ( γs ± , s ± δ ) = ( γt ± , t ± δ ). By the preceding discussion, this is so if and only if( λ s ± , ρ s ± ) = ( λ t ± , ρ t ± ), i.e., s ± = t ± . This leads us to consider the relation µ = µ S suchthat ( s, t ) ∈ µ if and only if(7.1) s + = t + , s − = t − , ( sp ) + = ( tp ) + and ( ps ) − = ( pt ) − for all p ∈ S + . RACE- AND PSEUDO-PRODUCTS 11
Remark 7.5.
We may write (7.1) as ( s, t ) ∈ µ if and only if ( ps ) ± = ( pt ) ± and ( sp ) ± = ( tp ) ± for all p ∈ S + since ( ps ) + = ps + = pρ s + by (4.1b), etc. Now we define a relation θ on S to be projection-separating if p, q ∈ S + and ( p, q ) ∈ θ imply that p = q . Clearly, µ is projection-separating. It may not be a congruence, butwe do have the following. Lemma 7.6.
Let θ be a ± -congruence on S . Then θ is projection-separating if and onlyif θ ⊆ µ .Proof. If ( s, t ) ∈ θ , then ( s ± , t ± ) ∈ θ , and since θ is projection-separating, s ± = t ± .Moreover, (( sp ) ± , ( tp ) ± ) ∈ θ for any p ∈ S + , whence ( sp ) ± = ( tp ) ± . Similarly ( ps ) ± =( pt ) ± , and ( s, t ) ∈ µ ensues by (7.1).Conversely, if θ ⊆ µ and ( s + , t + ) ∈ θ , then ( s + , t + ) ∈ µ , and by appropriate choices of p in equation (7.1) we have s + s + = t + s + = t + t + , whence s + = t + and θ is projection-separating. (cid:3) By analogy with the theory of inverse semigroups, we say that the localizable semi-group T is fundamental if µ T is the identity relation.8. The groupoid case: ∗ -localizable semigroups Recall that a category C is a groupoid if each morphism x has an inverse in C ,denoted by x − , such that x ◦ x − = x + and x − ◦ x = x − . (This is equivalent to C being cancellative and regular in the von Neumann sense.) Definition 8.1. A transcription groupoid is a groupoid which is also a transcriptioncategory. We shall provide a description of the corresponding transcription semigroup S ( C ),and to this end make the following definitions. Definitions 8.2. ( S, · , ∗ ) is a regular unary semigroup if it satisfies xx ∗ x = x and (8.1a) x ∗ xx ∗ = x ∗ (8.1b) for all x ∈ S . A regular unary semigroup ( S, · , ∗ ) is ∗ -localizable if it also satisfies ( x ( xx ) ∗ x ) ∗ = x ( xx ) ∗ x, (8.1c) x ( xyy ∗ ) ∗ = xy ( xy ) ∗ and (8.1d) ( x ∗ xy ) ∗ y = ( xy ) ∗ xy. (8.1e)The following observation is well known. Lemma 8.3.
Let ( S, · , ∗ ) be a regular unary semigroup. Then ∗ fixes every idempotentof S if and only if (8.1c) holds.Proof. If ∗ fixes every idempotent, then it certainly fixes x ( xx ) ∗ x since (8.1b) impliesthis is an idempotent for every x ∈ S . Conversely, if (8.1c) holds and if e = e , then e ∗ (8.1a) = ( ee ∗ e ) ∗ = ( e ( ee ) ∗ e ) ∗ (8.1c) = e ( ee ) ∗ e = ee ∗ e (8.1a) = e . (cid:3) Lemma 8.4.
Let ( S, · , ∗ ) be a regular unary semigroup satisfying (8.1d) and (8.1e) .Then the identity (8.2) x ∗∗ = x holds. If, in addition, (8.1c) holds, then ∗ fixes every e ∈ S satisfying ( e ∗ ) = e ∗ .Proof. (1) First we show x ∗ x ∗∗ = x ∗ x (8.3a) x ∗∗ x ∗ = xx ∗ . (8.3b)For (8.3a), we compute x ∗ x ∗∗ (8.1b) = x ∗ ( x ∗ xx ∗ ) ∗ (8.1d) = x ∗ x ( x ∗ x ) ∗ = x ∗ xx ∗ x (8.1a) = x ∗ x , using Lemma 8.3 in the third equality, and (8.3b) is proved dually. Thus x ∗∗ (8.1b) = x ∗∗ x ∗ x ∗∗ (8.3a) = x ∗∗ x ∗ x (8.3b) = xx ∗ x (8.1a) = x , which is (8.2).(3) If ( e ∗ ) = e ∗ , then Lemma 8.3 gives e ∗∗ = e ∗ and so e ∗ = e by (8.2). (cid:3) Since (8.1a) and (8.2) evidently imply (8.1b), we could equally well have used (8.2)as an axiom in place of (8.1b).
Theorem 8.5.
Let ( S, · , ∗ ) be a ∗ -localizable semigroup and define unary operations + , − : S → S by x + = xx ∗ and x − = x ∗ x for all x ∈ S . Then ( S, · , + , − ) is a localizablesemigroup.Conversely, let ( S, · , + , − ) be a localizable semigroup, assume that S is regular andsuppose there exists an inverse mapping ∗ : S → S ; x x ∗ such that x + = xx ∗ and x − = x ∗ x for all x ∈ S . Then ( S, · , ∗ ) is ∗ -localizable.Proof. ( ⇒ ) From (8.1a), we immediately have x + x = x = xx − , x + x + = x + , x − x − = x − .From (8.1b), we also get x ∗ x + = x ∗ and x − x ∗ = x ∗ .In (8.1d), replace x with xy + . The left hand side becomes xy + ( xy + y + ) ∗ = ( xy + ) + .The right side becomes xy + y ( xy + y ) ∗ = ( xy ) + . Therefore(8.4) ( xy + ) + = ( xy ) + . A dual argument using (8.1e) gives ( x − y ) − = ( xy ) − .Next we have(8.5) ( x + y ) + y = x + y . Indeed,( x + y ) + y = x + y ( x + y ) ∗ y = x + y (( x + ) − y ) ∗ y (8.1e) = x + y ( x + y ) ∗ x + y (8.1a) = x + y . RACE- AND PSEUDO-PRODUCTS 13
Now we compute ( x + y + ) ∗ ( x + y + ) ∗ (8.5) = ( x + y + ) ∗ (( x + y + ) + y + ) ∗ = ( x + y + ) ∗ ( x + y + ) + (( x + y + ) + y + ) ∗ (8.1d) = ( x + y + ) ∗ (( x + y + ) + y ) +(8.5) = ( x + y + ) ∗ ( x + y ) +(8.4) = ( x + y + ) ∗ ( x + y + ) + = ( x + y + ) ∗ . Thus by Lemma 8.4, we obtain ( x + y + ) ∗ = x + y + , (8.6)hence x + y + x + y + = x + y + . (8.7)Finally, ( x + y ) + (8.4) = ( x + y + ) + = x + y + ( x + y + ) ∗ (8.6) = x + y + x + y + (8.7) = x + y + . A dual argument gives ( xy − ) − = x − y − .For the converse, we have( xy ) + = ( xy + ) + = xy + ( xy + ) ∗ = xx − y + ( xy + ) ∗ = x ( xy + ) − ( xy + ) ∗ = x ( xy + ) ∗ , which is (8.1d), and (8.1e) is proved dually. By Lemma 8.4, we also have (8.2).Finally we show that ∗ fixes every idempotent. We first do this for elements of S + :( x + ) ∗ = ( x + ) ∗ x ++ = ( x + ) ∗ x + = ( x + ) − = x + . Now assume e = e . Then e + = ee ∗ = eee ∗ = ee + , and so e ∗ = e ∗ e + = e ∗ ee + = e − e + = ( ee + ) − = ( e + ) − = e + . This gives e (8.2) = e ∗∗ = ( e + ) ∗ = e + = e ∗ . By Lemma 8.3, (8.1c) holds, completing theproof. (cid:3) Hence when we consider ( S, · , ∗ ) as a localizable semigroup, we understand that itsprojections are given by x + = xx ∗ and x − = x ∗ x .Note that for the converse in Theorem 8.5, one must assume some compatibilitybetween an inverse mapping and the unary operations defining localizability; regularityof the semigroup is not sufficient. Consider, for instance, the 2-element semilattice S = { , } with 0 <
1. Then x ∗ = x is the only inverse mapping and xx ∗ = x ∗ x = x ,but x + = x − = 1 for all x ∈ S gives S the structure of a localizable semigroup. Corollary 8.6. ( S, · , ∗ ) is a ∗ -localizable semigroup if and only if C ( S ) is a transcriptiongroupoid. Proof.
Let S be a ∗ -localizable semigroup. By Theorem 4.6, C = C ( S ) is a transcriptioncategory. For any x ∈ C , x ◦ x ∗ is defined and equals xx ∗ = x + , and similarly x ∗ ◦ x = x − ,showing that C is a groupoid. Conversely, let C be a transcription groupoid; by Theorem4.4, S ( C ) is a localizable semigroup. Since it is also a groupoid, the map x x − satisfies the conditions for the converse part of Theorem 8.5 and so S is ∗ -localizable. (cid:3) The following result is Theorem 3 in [4]; for completeness, we give a short proof usingthe notation of the present work.
Proposition 8.7.
Let S be a ∗ -localizable semigroup. Then every idempotent is a pro-jection; in fact, for x ∈ S , x = x implies x = x + = x − . Consequently, S is orthodox.Proof. If x = x then xxx ∗ = xx ∗ and so xx + = x + . Then x + = ( x + ) − = ( xx + ) − = x − x + , by (4.1c) with y = x + . Dually or similarly, x − = x − x + ; and then x + = x − and x = xx − = xx + = x + as already seen. (cid:3) Remarks. (1) Proposition 8.7 contrasts with the general situation: if C is merely a category, E ( C ) = { x ∈ E ( S ) : x + = x − } . (As usual, we denote the set of idempotents in S or C by E ( S ) or E ( C ) respectively.) For let x ∈ S with x = x and x + = x − .Then by Definition 4.5, x ◦ x is defined in C and x ◦ x = x in C ( S ) = C . In theconverse direction, let x ◦ x = x in C . Then x + = x − and x = x ⊗ x = x .(2) The regular involution ∗ is not necessarily an anti-automorphism of S . Forexample, let S be a band and set x ∗ = x = x + = x − for each x ∈ S ; it is easilyverified that C = C ( S ) is a transcription groupoid. If S is not a semilattice thenthere are x, y ∈ S such that xy = yx , which is equivalent to ( xy ) ∗ = y ∗ x ∗ . References [1] Ara´ujo, J., Kinyon, M. K., Konieczny, J., Malheiro, A.: Four notions of conjugacy for abstractsemigroups. Proc. Roy. Soc. Edinburgh Sect. A. , 1169–1214 (2017)[2] Cockett, J. R. B., Lack, S.: Restriction categories I. Categories of partial maps. Theoret. Comput.Sci. , 223–259 (2002)[3] El-Qallali, A.: Congruences on ample semigroups. Semigroup Forum , 607–631 (2019) https://doi.org/10.1007/s00233-018-9988-4 [4] FitzGerald, D. G.: Groupoids on a skew lattice of objects. Art Discr. Appl. Math. https://doi.org/10.26493/2590-9770.1342.109 [5] Gould, V.: Notes on restriction semigroups. (2010). Accessed 22 August 2020[6] Gould, V., Stokes, T.: Constellations and their relationship with categories. Algebra Universalis , 271–304 (2017)[7] Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. , 21–57 (2009)[8] Hollings, C.: The Ehresmann-Schein-Nambooripad theorem and its successors. Eur. J. Pure Appl.Math. , 414–450 (2012)[9] Jones, P. R.: A common framework for restriction semigroups and regular ∗ -semigroups. J. PureAppl. Algebra , 618–632 (2012)[10] Kudryavtseva, G.: Two-sided expansions of monoids. Internat. J. Algebra Comput. , 1467–1498(2019)[11] Lawson, M. V. Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, RiverEdge, NJ (1998) RACE- AND PSEUDO-PRODUCTS 15 [12] Lawson, M. V.: Semigroups and ordered categories I. The reduced case. J. Algebra , 422–462(1991)[13] Malandro, M.E.: Enumeration of finite inverse semigroups. Semigroup Forum , 679–729 (2019) https://doi.org/10.1007/s00233-019-10054-9 [14] Miller, D. D., Clifford, A. H.: Regular D -classes in semigroups. Trans. Amer. Math. Soc. ,270–280 (1956)[15] Nordahl, T. E., Scheiblich, H. E.: Regular ∗ -semigroups. Semigroup Forum , 369–377 (1978)[16] Petrich, M., Reilly, N. R.: Completely Regular Semigroups, Canadian Mathematical Society Seriesof Monographs and Advanced Texts, Vol. 23. Wiley, New York, NY (1999)[17] Stokes, T.: D-semigroups and constellations. Semigroup Forum , 442–462 (2017)[18] Stokes, T.: Generalized domain and E-inverse semigroups. Semigroup Forum , 32–52 (2018)[19] Stokes, T.: How to generalise demonic composition. Semigroup Forum (2020) https://doi.org/10.1007/s00233-020-10117-2 [20] Szendrei, M.B.: Structure theory of regular semigroups. Semigroup Forum , 119–140 (2020) https://doi.org/10.1007/s00233-019-10055-8https://doi.org/10.1007/s00233-019-10055-8