aa r X i v : . [ m a t h . C T ] F e b FINITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS
MARK V. LAWSON
Abstract.
We describe the structure of finite Boolean inverse monoids andapply our results to the representation theory of finite inverse semigroups. Wethen generalize to semisimple Boolean inverse semigroups. Introduction
The goal of this paper is to prove a number of theorems about finite and semisim-ple Boolean inverse semigroups. Although semisimple Boolean inverse semigroupsinclude finite Boolean inverse monoids as a special case, we treat them separatelysince the finite case can be handled using only elementary means.It turns out that the theory of finite Boolean inverse monoids, described inSection 4, is tightly intertwined with the theory of finite groupoids. In fact, thetheory of finite Boolean inverse monoids generalizes the theory of finite Booleanalgebras in exactly the same way as the theory of finite groupoids generalizes thetheory of finite sets: specifically, we prove in Theorem 4.5 that every finite Booleaninverse monoid is isomorphic to the Boolean inverse monoid of all local bisectionsof a finite discrete groupoid: namely, the finite discrete groupoid of its atoms; inTheorem 4.7, we prove that every finite Boolean inverse monoid is isomorphic toa finite direct product of 0-simplifying Boolean inverse monoids each of which isisomorphic to all the finite square rook matrices over a finite group with a zeroadjoined; in Theorem 4.11, we prove that the Booleanization of a finite inversesemigroup (without zero) S is isomorphic to the Boolean inverse monoid of all localbisections of the set S equipped with the restricted product. We apply some of ourresults on the finite case to the study of the representation theory of finite inversesemigroups.The theory of semisimple Boolean inverse semigroups, described in Section 5,generalizes the theory of finite Boolean inverse monoids but we use non-commutativeStone duality to achieve our goals. In Theorem 5.7, we prove that the semisimpleBoolean inverse semigroups are precisely those in which the associated groupoid ofprime filters is endowed with the discrete topology; this leads to our characteriza-tion of semisimple Boolean inverse semigroups in Corollary 5.8 as being isomorphicto the Boolean inverse semigroup of all finite local bisections of a discrete groupoid;in Theorem 5.14, we prove that every semisimple Boolean inverse semigroup is iso-morphic to a restricted direct product of 0-simplifying Boolean inverse semigroupseach of which is isomorphic to all the square rook matrices over a group with zeroadjoined. But it is in Theorem 5.15, the Dichotomy Theorem, in which the rˆole ofsemisimple Boolean inverse semigroups within the wider structure theory of Booleaninverse semigroups is made clear: we prove that every 0-simplifying Boolean inversesemigroup is either semisimple or atomless. A countable atomless Boolean inversemonoid is called a Tarski monoid . The theory of such monoids is discussed in [8, 9].In Theorem 5.16, we prove that an inverse semigroup is semisimple if and only ifits universal groupoid is discrete.
I would like to thank David Janin of the Universit´e de Bordeaux for the opportunity to visithim in April 2018 where much of the work for this paper was carried out.
The type monoids of semisimple Boolean inverse semigroups are characterizedin Section 6. Sections 2 and 3 contain background results we need.2.
Background results
Posets.
Let ( X, ≤ ) be a poset. If A ⊆ X then define A ↑ = { x ∈ X : a ≤ x for some a ∈ A } and A ↓ = { x ∈ X : x ≤ a for some a ∈ A } . If A = { a } then we write a ↑ = { a } ↑ and a ↓ = { a } ↓ . If A = A ↓ we say that A is an order-ideal . If A is a singleton set then A ↓ is called a principal order-ideal .2.2. Groupoids.
See the book by Higgins [3] for general groupoid theory. For usa groupoid is a small category in which every element is invertible. We denote theunique inverse of the element x by x − . If G is a groupoid we write d ( x ) = x − x ,for the domain of x , and r ( x ) = xx − , for the range of x . Observe that for uscategories are ‘one-sorted structures’ and so we identify the objects of the categorywith the identities. The set of identities of G is denoted by G o . If e is an identity,define G e to be all elements g such that g − g = e = gg − . Then G e is a groupcalled the local group at e . A groupoid is said to be principal if all local groups aretrivial. A principal groupoid is just an equivalence relation viewed as a groupoid.If e and f are identities we write e D f if there is an element of the groupoid whosedomain is e and whose range is f . If g and h are elements of the groupoid wewrite g D h if d ( g ) D d ( h ). The relation D is an equivalence relation on G whoseequivalence classes are called the connected components of G . A groupoid is saidto be connected if it has exactly one connected component. Every groupoid is adisjoint union of its connected components each of which is a connected groupoid.The proof of the following is well-known. Lemma 2.1.
Let H be a group and X a non-empty set. Then X × H × X is aconnected groupoid when the partial multiplication is defined by ( x, g, y )( y, h, z ) =( x, gh, z ) . Here, d ( x, g, y ) = ( y, , y ) , r ( x, g, y ) = ( x, , x ) and ( x, g, y ) = ( y, g − , x ) .The identities are the elements ( x, , x ) . Every connected groupoid is isomorphic toa groupoid of this form. Let G be a groupoid. A subset A ⊆ G is called a local bisection if a, b ∈ A and d ( a ) = d ( b ) (respectively, r ( a ) = r ( b )) implies that a = b . The set-theoreticproduct of two local bisections is a local bisection. Denote by K ( G ) the set of alllocal bisections of G . Denote by K fin ( G ) the set of all finite local bisections of G .2.3. Inverse semigroups.
For background results in inverse semigroup theory,we shall refer to [6]. If S is an inverse semigroup, we denote its semilattice ofidempotents by E ( S ). If X ⊆ S then define E ( X ) = X ∩ E ( S ). If I ⊆ S then I is a (semigroup) ideal if SI, IS ⊆ I . One issue that will haunt this paper is thepresence or absence of a zero since it affects morphisms. Thus homomorphismsbetween inverse semigroups with zero are required to preserve zero. If S is aninverse semigroup with a zero then S ∗ = S \ { } . If S is an inverse semigroupwithout a zero then we can adjoin one to get S , an inverse semigroup with zero.If θ is a homomorphism with domain S then the corresponding congruence in-duced on S is denoted by cong( θ ).An inverse semigroup is said to be fundamental if the only elements of the semi-group commuting with all idempotents are themselves idempotents. Equivalently, INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 3 an inverse semigroup is fundamental if the maximum idempotent-separating con-gruence µ is trivial [6, Pages 139, 140]. Fundamental inverse semigroups are ofsignal importance in the general theory of inverse semigroups; see [6, Section 5.2].If a is an element of an inverse semigroup S then define d ( a ) = a − a and r ( a ) = aa − ; we also write d ( a ) a −→ r ( a ) and a D b . The restricted product a · b ofthe elements a and b in an inverse semigroup is defined precisely when d ( a ) = r ( b )in which case it is equal to ab . With respect to the restricted product, S becomesa groupoid which we denote by S · [6, Proposition 3.1.4]. Observe that if a and b are both nonzero and d ( a ) = r ( b ) then a · b is also nonzero. The following thereforemakes sense: define G ( S ) to be the groupoid S · if S does not have a zero and thegroupoid with set S ∗ equipped with the restricted product if S does have a zero.We call G ( S ) the restricted groupoid of S .The natural partial order is defined by a ≤ b if and only if a = ba − a . An inversesemigroup is partially ordered with respect to the natural partial order. An inversesemigroup with all binary meets with respect to the natural partial order is called a meet-semigroup or ∧ -semigroup . The proof of the following is straightforward fromthe properties of the natural partial order. Lemma 2.2.
Let S be an inverse semigroup. If x, y ≤ a and d ( x ) = d ( y ) (respec-tively, r ( x ) = r ( y ) ) then x = y . An element a of an inverse semigroup is called an atom if x ≤ a implies x = a or x = 0. An inverse semigroup is said to be atomless if it has no atoms. Lemma 2.3.
Let S be an inverse semigroup in which a D b . Then a is an atom ifand only if b is an atom.Proof. We prove that a an atom implies that b is an atom. The result then followsby symmetry. Let d ( b ) x → d ( a ). Suppose that c ≤ b . Then a r ( x d ( c )) ≤ a . But a is an atom. It follows that a r ( x d ( c )) = 0 or a r ( x d ( c )) = a . If the former, then r ( x d ( x )) = 0 and so x d ( c ) = 0 which yields d ( c ) = 0 and so c = 0. If the latterthen d ( a ) = r ( x d ( c )). It follows that d ( c ) = d ( x ) = d ( b ). Thus c = b . We haveaccordingly proved that b is an atom. (cid:3) The compatibility relation is defined by a ∼ b if and only if a − b and ab − areboth idempotents; the orthogonality relation is defined by a ⊥ b if and only if a − b = 0 and ab − = 0. A necessary condition for two elements of an inversesemigroup to have a join is that they be compatible. A non-empty subset of aninverse semigroup is said to be compatible if each pair of elements of the subset iscompatible. The following is [6, Lemma 1.4.11, Lemma 1.4.12]. Lemma 2.4.
Let S be an inverse semigroup. (1) a ∼ b if and only if a ∧ b exists and d ( a ∧ b ) = d ( a ) d ( b ) and r ( a ∧ b ) = r ( a ) r ( b ) . (2) If a ∼ b then a ∧ b = ab − b = bb − a = aa − b = ba − a . The following was proved as [6, Proposition 1.4.19].
Lemma 2.5.
Let S be an inverse semigroup, let A be a non-empty subset and let s ∈ S . (1) If V A exists then V a ∈ A sa exists and s ( V A ) = V a ∈ A sa . (2) If V A exists then V a ∈ A as exists and ( V A ) s = V a ∈ A as . The proof of the following is straightforward.
Lemma 2.6.
Let S be an inverse semigroup. (1) a ⊥ b implies that ca ⊥ cb for any c ∈ S . MARK V. LAWSON (2) a ⊥ b implies that ac ⊥ bc for any c ∈ S . Lemma 2.7.
In an inverse semigroup, let a ∼ b . Then the following are equivalent: (1) a ⊥ b . (2) d ( a ) ⊥ d ( b ) . (3) r ( a ) ⊥ r ( b ) .Proof. By symmetry, it is enough to prove the equivalence of (1) and (2). Clearly,(1) implies (2). We prove that (2) implies (1). Since a ∼ b we have that a − b isan idempotent. Hence aa − bb − ≤ ab − . But ab − = a ( a − ab − b ) b − = 0. Thus aa − bb − ≤ ab − = 0. We have proved that r ( a ) ⊥ r ( b ) and so a ⊥ b . (cid:3) We shall need the following properties of the restricted product.
Lemma 2.8.
Let S be an inverse semigroup. (1) Let a, b ∈ S . Then ab = a ′ · b ′ where a ′ ≤ a and b ′ ≤ b . (2) Suppose that ab is not a restricted product. Then ab = a b , where a < a ,or ab = ab , where b < b . (3) Let x ≤ a · b . Then x = a ′ · b ′ where a ′ ≤ a and b ′ ≤ b . (4) Let a, b ∈ S . Then ( a ↓ )( b ↓ ) = ( ab ) ↓ , where on the left we use products ofsubsets.Proof. (1) Put a ′ = abb − and b ′ = a − ab . It is now routine to check that d ( a ′ ) = r ( b ) and that ab = a ′ · b ′ .(2) We are given that a − a = bb − . Suppose that abb − = a and a − ab = b .Then a − abb − = a − a and so a − a ≤ bb − and a − abb − = bb − and so bb − ≤ a − a . From which it follows that a − a = bb − which is a contradiction. It followsthat either abb − = a or a − ab = b . The result is now clear since abb − ≤ a and a − ab ≤ b .(3) We have that x = a ( b d ( x )). Thus by (1) above, we have that x = a ′ · b ′ where a ′ ≤ a and b ′ ≤ b d ( x ) ≤ b .(4) The fact that ( a ↓ )( b ↓ ) ⊆ ( ab ) ↓ is immediate from the fact that inverse semi-groups are partially ordered with respect to the natural partial order. The reverseinclusion follows by (3). (cid:3) Boolean inverse semigroups
In this section, we shall describe the properties of Boolean inverse semigroupswe shall need. We refer the reader to [22] for the general theory of Boolean inversesemigroups.3.1.
Lattices. A distributive lattice has a bottom element but not necessarily atop element; if it does have a top, it is said to be unital . For the theory of Booleanalgebras, we refer to [1] but we highlight some non-standard terminology here.We use the term unital Boolean algebra to mean what is usually defined to besimply a ‘Boolean algebra’ and Boolean algebra to mean what is usually termeda ‘generalized Boolean algebra’. Specifically, a Boolean algebra is a distributivelattice that is locally a unital Boolean algebra — this means that for each element x in the lattice the subset x ↓ of elements beneath x is a unital Boolean algebra.In a Boolean algebra, if e ≤ f then there is a unique element, denoted by ( f \ e ),such that f = e ∨ ( f \ e ) and e ∧ ( f \ e ) = 0. The element ( f \ e ) is the relativecomplement . Observe that ( f \ e ) is the unique element g such that g ≤ f suchthat g ∧ e = 0 and f = e ∨ g . In a unital Boolean algebra, we denote the (absolute)complement of an element e by ¯ e ; as usual, we have that 1 = e ∨ ¯ e and e ∧ ¯ e = 0.Observe that e ≤ f if and only if e ¯ f = 0. If X is any set then P ( X ) denotes the powerset of X . This is a unital Boolean algebra for subset inclusion. If X is an INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 5 infinite set then P fin ( X ) denote the set of all finite subsets of X . This is a Booleanalgebra.3.2. Definition of Boolean inverse semigroups.
An inverse semigroup is saidto be distributive if compatible elements have joins and multiplication distributesover such joins. A distributive inverse semigroup is said to be
Boolean if the semi-lattice of idempotents forms a Boolean algebra.
Remark 3.1.
We shall assume throughout this paper that our Boolean inversesemigroups are not just the zero semigroup.If a ⊥ b then a ∨ b is often written as a ⊕ b and is called an orthogonal join . Thefollowing class of examples of Boolean inverse semigroups is fundamental to thispaper. Proposition 3.2.
Let G be a discrete groupoid. Then K fin ( G ) , the set of all finitelocal bisections of G , is a Boolean inverse semigroup.Proof. The set of all local bisections of G is a pseudogroup [11, Proposition 2.1].The product of two finite local bisections is a finite local bisection. Observe thatthe inverse of A is A − . It follows that K fin ( G ) is a distributive inverse semigroup.The finite local bisections in the set of identities of G are the finite subsets of theset of identities. Thus the poset of idempotents of K fin ( G ) is order-isomorphic tothe set of all finite subsets of the set of identities of G . Thus K fin ( G ) is a Booleaninverse semigroup. (cid:3) The following two results were proved as [8, Lemma 2.5(3)] and [8, Lemma 2.5(4)],respectively.
Lemma 3.3.
Let S be a distributive inverse semigroup. (1) Suppose that W mi =1 a i and c ∧ ( W mi =1 a i ) both exist. Then all meets c ∧ a i exist as does the join W mi =1 ( a i ∧ c ) and we have that c ∧ m _ i =1 a i ! = m _ i =1 ( a i ∧ c ) . (2) Suppose that a ∈ S and b = W nj =1 b j are such that all meets a ∧ b j exist.Then a ∧ b exists and is equal to W nj =1 a ∧ b j . If y ≤ x then d ( y ) ≤ d ( x ) and so the following definition makes sense:( x \ y ) = x ( d ( x ) \ d ( y )) . Lemma 3.4.
Let S be a Boolean inverse semigroup. (1) d ( x \ y ) = ( d ( x ) \ d ( y )) . (2) If y ≤ x then y ⊥ ( x \ y ) and x = y ∨ ( x \ y ) . (3) Suppose that a ≤ x is such that a ⊥ y and x = y ∨ a then a = ( x \ y ) . (4) r ( x \ y ) = ( r ( x ) \ r ( y )) .Proof. (1) Straightforward.(2) Observe that ( x \ y ) ≤ x and that d ( x \ y ) = d ( x ) \ d ( y ). It follows that x ∼ ( x \ y ) and so, by Lemma 2.7, we have that y ⊥ ( x \ y ). Clearly, y ∨ ( x \ y ) ≤ x .But d ( y ∨ ( x \ y )) = d ( y ) ∨ ( d ( x ) \ d ( y )) = d ( x ). It follows that x = y ∨ ( x \ y ) ≤ x .(3) Observe that d ( a ) ≤ d ( x ), d ( a ) ⊥ d ( y ) and d ( x ) = d ( y ) ∨ d ( a ). Thus in theBoolean algebra, we have that d ( a ) ≤ d ( x ) \ d ( y ). It follows that a ≤ ( x \ y ).(4) This follows by part (3) above. (cid:3) The following was proved as [10, Lemma 2.2] and [22, Lemma 3.1.12].
Lemma 3.5.
Let S be a Boolean inverse semigroup. MARK V. LAWSON (1) ( s \ t ) = s − \ t − . (2) a ( s \ t ) = as \ at . and ( s \ t ) a = sa \ ta . (3) s ( u ∨ v ) = ( s \ u ) s − ( s \ v ) . (4) ( s \ t )( u \ v ) = su \ ( sv ∨ tu ) . (5) a ≤ b ≤ c implies that c \ c ≤ c \ a . Lemma 3.6.
Let S be a Boolean inverse semigroup. Then every binary join isequal to an orthogonal join.Proof. Let x ∼ y so that x ∨ y is defined. By Lemma 2.4, the meet x ∧ y exists and d ( x ∧ y ) = d ( x ) d ( y ) and r ( x ∧ y ) = r ( x ) r ( y ). Since x ∧ y ≤ x we can construct theelement x \ ( x ∧ y ). Thus x ∨ y = ( x ∧ y ) ∨ ( x \ ( x ∧ y )) ∨ y . But x ∧ y ≤ y . It followsthat x ∨ y = ( x \ ( x ∧ y )) ∨ y . Clearly, y ∼ ( x \ ( x ∧ y )) and d ( y ) ⊥ d ( x \ ( x ∧ y )).It follows by Lemma 2.7 that y ⊥ x \ ( x ∧ y ). (cid:3) Although we have only talked above about compatible binary joins and orthog-onal binary joins, our results extend to n -ary joins of either type where n is anynon-zero natural number. Lemma 3.7.
Let s = W mi =1 s i where the s i are distinct and non-zero. Then thereis an orthogonal set of elements { t , . . . , t m } such that (1) s = W mi =1 t i . (2) For each i , we have that t i ≤ s i .Proof. The set { s , . . . , s m } is a compatible one. Thus all the joins s , s ∨ s , s ∨ s ∨ s , . . . exist. In addition, by Lemma 2.4, all the meets s ∧ s , ( s ∨ s ) ∧ s ,( s ∨ s ∨ s ) ∧ s , . . . exist. Define t = s and t i = s i \ (( s ∨ . . . s i − ) ∧ s i )for each i = 2 , . . . , m . Observe that { t , . . . , t m } is also a compatible subset.Now, d ( t ) = d ( s ) and for i ≥ d ( t i ) = d ( s i ) \ d (( s ∨ . . . ∨ s i − ) ∧ s i ). In particular, d ( t i ) ≤ d ( s i ). Using Lemma 2.7, we have that d ( t i ) ⊥ d ( s ) , . . . , d ( s i − ). It is therefore clear that t is orthogonal to t i when i ≥ t i is orthogonal to t j when i < j . Observe that s i = t i ⊕ (( s ∨ . . . ∨ s i − ) ∧ s i )Clearly, W mi =1 t i ≤ s . Now observe that t = s , t ∨ t = s ∨ s by Lemma 3.6.Assume that t ∨ . . . ∨ t i − = s ∨ . . . ∨ s i − . Then t ∨ . . . ∨ t i = ( s ∨ . . . ∨ s i − ) ∨ t i = ( s ∨ . . . ∨ s i − ) ∨ ( s i \ (( s ∨ . . . s i − ) ∧ s i )))which is equal to s ∨ . . . ∨ s i . (cid:3) The following result is frequently invoked in proofs.
Proposition 3.8.
The following are equivalent. (1) S is a Boolean inverse semigroup. (2) S has all binary orthogonal joins, multiplication distributes over binaryorthogonal joins, and its semilattice of idempotents forms a Boolean algebrawith respect to the natural partial order.Proof. It is enough to prove that (2) ⇒ (1). We prove first that S has all binarycompatible joins. Let a, b ∈ S such that a ∼ b . Then, by Lemma 2.4, we knowthat a ∧ b exists. The idempotent d ( a ) \ d ( a ) d ( b ) is defined since E ( S ) is a Booleanalgebra. Define a ′ = a ( d ( a ) \ d ( a ) d ( b )). Then a ′ and a ∧ b are both less than a and compatible. In addition, d ( a ′ ) ⊥ d ( a ∧ b ). Thus by Lemma 2.7, we have that a ′ ⊕ ( a ∧ b ) is defined. It is clear that a = a ′ ⊕ ( a ∧ b ). Similarly, b ′ = b ( d ( b ) \ d ( a ) d ( b ))is less than or equal to b and b = b ′ ⊕ ( a ∧ b ). The elements a ′ , ( a ∧ b ) , b ′ are pairwiseothogonal. We have that a ∨ b = ( a ′ ⊕ ( a ∧ b )) ⊕ b ′ . Thus a ∨ b is defined. Nowlet c be any element. We prove that c ( a ∨ b ) = ca ∨ cb . We use Lemma 2.6, so INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 7 that if a ⊥ b then ca ⊥ cb and c ( a ⊕ b ) = ca ⊕ cb by assumption. Observe that c (( a ′ ⊕ ( a ∧ b )) ⊕ b ′ ) = ( ca ′ ⊕ c ( a ∧ b )) ⊕ cb ′ , by assumption. But c ( a ∧ b ) = ca ∧ cb by Lemma 2.5. Thus ca ′ = ca ( d ( ca ) \ d ( ca ) d ( cb )) and cb ′ = cb ( d ( cb ) \ d ( ca ) d ( cb ))where we use the fact that multiplication distributes over orthogonal joins. Theresult now follows. (cid:3) An additive homomorphism θ : S → T between Boolean inverse semigroups is asemigroup homomorphism that maps zero to zero and preserves binary compatiblejoins. Observe that an additive homomorphism θ induces a map from E ( S ) to E ( T )that preserves meets and joins; in addition, if f ≤ e then θ ( e \ f ) = θ ( e ) \ θ ( f ).If S and T are also both monoids then we say that θ is unital if it also mapsthe identity to the identity. In what follows, we shall usually just write morphism rather than additive homomorphism for brevity.3.3. Additive ideals.
Let S be a Boolean inverse semigroup. If A ⊆ S define A ∨ to be the set of all binary compatible joins of elements of A . An additive ideal I ina Boolean inverse semigroup is a semigroup ideal I such that if a, b ∈ I and a ∼ b then a ∨ b ∈ I . Lemma 3.9.
Let S be a Boolean inverse semigroup. Let A ⊆ S be a non-emptysubset. Then the smallest additive ideal containing A is I = ( SAS ) ∨ .Proof. Clearly, A ⊆ I . We check first that I is an additive ideal. Let a ∈ I and s ∈ S . Then a = W mi =1 a i where a i ∈ SAS . We have that sa = W mi =1 sa i . But,clearly, sa i ∈ SAS . It follows that sa ∈ I . A symmetric argument shows that sa ∈ I . We have therefore proved that I is a (semigroup) ideal. Let { a , . . . , a m } be a compatible subset of I . Then, each a i is a join of elements of SAS . Thus W mi =1 a i is a join of elements of SAS . It follows that I is an additive ideal. Now let J be any additive ideal containing A . Then since A ⊆ J we have that SAS ⊆ J .But J is additive and so ( SAS ) ∨ ⊆ J . We have therefore proved that I ⊆ J . (cid:3) Both { } and S are additive ideals. If these are the only additive ideals and S = { } then we say that S is 0 -simplifying . The following relation was introducedin [13]. Let e and f be two non-zero idempotents in S . Define e (cid:22) f if andonly if there exists a set of elements X = { x , . . . , x m } such that e = W mi =1 d ( x i )and r ( x i ) ≤ f for 1 ≤ i ≤ m . We can write this informally as e = W d ( X ) and W r ( X ) ≤ f . We say that X is a pencil from e to f . The relation (cid:22) is a preorderon the set of idempotents. Lemma 3.10.
Let S be a Boolean inverse semigroup. Then f (cid:22) e if and only ifwhenever e ∈ I , an additive ideal, then f ∈ I .Proof. Suppose that f (cid:22) e and e ∈ I , an additive ideal. Then there is a pencil X from f to e . Thus e = W x ∈ X d ( x ) and r ( x ) ≤ f for 1 ≤ i ≤ m . Now f ∈ I implies that r ( x ) ∈ I since I is an order-ideal. It follows that d ( x ) ∈ I since I is asemigroup ideal. But I is closed under compatible joins and so f ∈ I , as required.We now prove the converse. The smallest additive ideal containing e is ( SeS ) ∨ byLemma 3.9. By assumption f ∈ ( SeS ) ∨ . Thus f = W mi =1 e i where e , . . . , e m ∈ SeS .But it is easy to prove that e i = x − i x i , where x i ∈ eSe i Thus x i x − i ≤ e . It followsthat X = { x , . . . , x m } is a pencil from f to e and so f (cid:22) e . (cid:3) Define the equivalence relation e ≡ f if and only if e (cid:22) f and f (cid:22) e . The follow-ing was proved as part of [13, Lemma 7.8] but is also immediate by Lemma 3.10. Lemma 3.11.
Let S be a Boolean inverse semigroup. Then ≡ is the universalrelation on the set of non-zero idempotents if and only if S is -simplifying. MARK V. LAWSON
The kernel of a morphism θ : S → T between two Boolean inverse semigroups,denoted by ker( θ ), is the inverse image under θ of the zero of T . Remark 3.12.
Readers familiar with semigroup theory are warned that our defi-nition of ‘kernel’ is not the one usual in semigroup theory.The proof of the following is straightforward.
Lemma 3.13.
The kernels of morphisms between Boolean inverse semigroups areadditive ideals.
Lemma 3.13 immediately raises the question of what can be said about mor-phisms whose kernels are trivial.
Lemma 3.14.
Let θ : S → T be a morphism between Boolean inverse semigroups.Then θ has a trivial kernel if and only if it is idempotent-separating.Proof. Suppose first that θ is idempotent-separating. Let θ ( a ) = 0. Then θ ( a − a ) =0. It follows by assumption that a − a = 0 and so a = 0 which implies that the θ is trivial. Conversely, suppose that the kernel of θ is trivial. We prove that θ is idempotent-separating. Let θ ( e ) = θ ( f ) where e and f are idempotents. Then θ ( e ) = θ ( e ∧ f ) = θ ( f ) since θ restricts to a morphism of Boolean algebras. Butthen θ ( e \ ( e ∧ f )) = 0 implies that e = e ∧ f . Similarly f = e ∧ f . Thus e = f , asrequired. (cid:3) Additive congruences.
The goal of this subsection is to define what wemean by a ‘simple’ Boolean inverse semigroup.We say that a congruence σ on a Boolean inverse semigroup S is additive if S/σ is a Boolean inverse semigroup and the natural map from S to S/σ is a morphism.The treatment of additive congruences in [22] cannot be bettered, and I shall simplysummarize the main definitions and results from there below. It is easy to checkthat a congruence is additive precisely when it preserves the operations ⊖ and ▽ introduced in [22, Page 82]. The following is [22, Proposition 3-4.1]. Proposition 3.15.
A congruence σ on a Boolean inverse semigroup S is additiveif and only if for all a ∈ S and orthogonal idempotents e and f we have that ae σ e and af σ f implies that a ( e ∨ f ) σ ( e ∨ f ) . Observe that [22, Example 3-4.2] shows that not all idempotent-separating homo-morphisms between Boolean inverse semigroups need be morphisms. The followingwas proved as [22, Proposition 3-4.5].
Proposition 3.16.
Let S be a Boolean inverse semigroup. Then S/µ is a Booleaninverse semigroup and the natural map S → S/µ is a morphism of Boolean inversesemigroups.
The following is [22, Proposition 3-4.6].
Proposition 3.17.
Let S be a Boolean inverse semigroup. Let I be an additiveideal of S . (1) Define ( a, b ) ∈ ε I if and only if there exists c ≤ a, b such that a \ c, b \ c ∈ I .Then ε I is an additive congruence with kernel I . (2) If σ is any additive congruence with kernel I then ε I ⊆ σ . An additive congruence is ideal-induced if it equals ε I for some additive ideal I .Let S and T be Boolean inverse semigroups. A morphism θ : S → T is said to be weakly-meet-preserving if for any a, b ∈ S and any t ∈ T if t ≤ θ ( a ) , θ ( b ) then thereexists c ≤ a, b such that t ≤ θ ( c ). Such morphisms were introduced in [11]. Thefollowing result is due to Ganna Kudryavtseva (private communication). INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 9
Proposition 3.18.
A morphism of Boolean inverse semigroups is weakly-meet-pre-serving if and only if its associated congruence is ideal-induced.Proof.
Let I be an additive ideal of S and let ε I be its associated additive con-gruence on S . Denote by ν : S → S/ε I is associated natural morphism. We provethat ν is weakly meet preserving. Denote the ε I -class containing s by [ s ]. Let[ t ] ≤ [ a ] , [ b ]. Then [ t ] = [ at − t ] and [ t ] = [ bt − t ]. By definition there exist u, v ∈ S such that u ≤ t, at − t and v ≤ t, bt − t such that t \ u, at − t \ u, t \ v, bt − t \ v ∈ I .Now [ t ] = [ u ] = [ at − t ] and [ t ] = [ v ] = [ bt − t ]. Since u, v ≤ t it follows that u ∼ v and so u ∧ v exists by Lemma 2.4. Clearly, u ∧ v ≤ a, b . In addition [ t ] = [ u ∧ v ].We have proved that ν is weakly-meet-preserving.Conversely, let θ : S → T be weakly-meet-preserving. We prove that it is de-termined by its kernel I . By part (2) of Proposition 3.17, it is enough to provethat if θ ( a ) = θ ( b ) then we can find c ≤ a, b such that a \ c, b \ c ∈ I . Put t = θ ( a ) = θ ( b ). Then there exists c ≤ a, b such that t ≤ θ ( c ). It is easy to checkthat θ ( a \ c ) = 0 = θ ( b \ c ). We have therefore proved that a \ c, b \ c ∈ I and so( a, b ) ∈ ε I . (cid:3) Most of the following is [22, Proposition 3-4.9].
Proposition 3.19 (Factorization of Boolean morphisms) . Let θ : S → T be amorphism of Boolean inverse semigroups. Let the kernel of θ be I . Put ε = ε I andlet ν : S → S/ε be the natural map. Then there is a unique morphism φ : S/ε → T such that φν = θ where φ is idempotent-separating.Proof. Observe first that ε ⊆ cong( θ ); suppose that ( a, b ) ∈ ε . Then there exists c ≤ a, b such that a \ c, b \ c ∈ I . We have that, θ ( a ) = θ (( a \ c ) ∨ c ) = θ ( c )and, by symmetry, θ ( b ) = θ ( c ). It follows that ( a, b ) ∈ cong( θ ). The function φ : S/ε → T defined by φ ([ a ]) = θ ( a ), where [ a ] denotes the ε -class of a , is a well-defined homomorphism. We prove that φ is idempotent-separating. Let φ ([ e ]) = φ ([ f ]) where e and f are idempotents. Then θ ( e ) = θ ( f ). Thus θ ( e \ ef ) = 0 and θ ( f \ ef ) = 0. We have shown that ( e, f ) ∈ ε and so [ e ] = [ f ]. (cid:3) Proposition 3.19 is included for interest.A Boolean inverse semigroup is simple if it has no non-trivial additive congru-ences.
Proposition 3.20.
A Boolean inverse semigroup is simple if and only if it is -simplifying and fundamental.Proof. Let S be a 0-simplifying fundamental Boolean inverse semigroup. Let σ beany Boolean congruence on S . If σ (0) = S then σ is the universal congruence.If σ = S then, since S is 0-simplifying, σ (0) = { } . Thus by Lemma 3.14, thecongruence σ is idempotent-separating. It follows that σ ⊆ µ . But S is fundamentaland so µ is the equality congruence. It follows that σ is equality. We have provedthat S is additive congruence-free. Conversely, suppose now that S is additivecongruence-free. Let I be a ∨ -ideal such that I = { } . Then by Proposition 3.17,we have that ε I is an additive congruence with kernel I . It follows that ε I is theuniversal congruence and so I = S . We have proved that S is 0-simplifying. ByLemma 3.16, the congruence µ is Boolean. Thus by assumption µ is equality andso S is fundamental. (cid:3) Proposition 3.20 explains why in what follows we are so interested in determiningwhen a Boolean inverse semigroup is 0-simplifying or fundamental.
Generalized rook matrices.
Boolean inverse semigroups are more ‘ring-like’ than arbitrary inverse semigroups. One way this is demonstrated is that wemay define matrices over such semigroups. The following construction was firstdescribed in [4] and then generalized in [5] and [21]. Let S be an arbitrary Booleaninverse semigroup and let X be a non-empty set. An | X | × | X | generalized rookmatrix over S is an | X | × | X | matrix with entries from S that satisfies the followingthree conditions:(RM1): If a and b are in distinct columns and lie in the same row of A then a − b = 0. That is r ( a ) ⊥ r ( b ).(RM2): If a and b are in distinct rows and lie in the same column of A then ab − = 0. That is d ( a ) ⊥ d ( b ).(RM3): In the case where | X | is infinite we also require that only a finitenumber of entries in the matrix are non-zero.We shall usually just say rook matrix instead of generalized rook matrix . When n is finite, we use the notation I n to mean the n × n identity matrix. Let A and B be rook matrices of the same size. Then the matrix AB is defined as follows:( AB ) ij = M k a ik b kj . The following is [5, Proposition 3.3].
Proposition 3.21.
Let S be a Boolean inverse semigroup. (1) If A and B are rook matrices such that AB is defined then AB is well-defined and is a rook matrix. (2) Multiplication is associative when defined. (3)
The matrices I n are identities when multiplication is defined. (4) Let A = ( a ij ) be a rook matrix. Define A ∗ = ( a − ji ) . Then A ∗ is a rookmatrix and A = AA ∗ A and A ∗ = A ∗ AA ∗ . (5) The idempotents are those square rook matrices which are diagonal andwhose diagonal entries are themselves idempotents. (6) If A and B are two rook matrices of the same size then A ≤ B if and onlyif a ij ≤ b ij for all i and j . (7) If A and B are again of the same size then A ⊥ B if and only if a ij ⊥ b ij ;if this is the case then A ∨ B exists and its elements are a ij ∨ b ij . Proposition 3.21 tells us, in particular, how to manipulate rook matrices overBoolean inverse semigroups. The set of all | X |×| X | rook matrices over S is denotedby M | X | ( S ). If X is finite then | X | = n , and we denote the corresponding set ofrook matrices by M n ( S ), and if X is countably infinite then | X | = ω , and we denotethe corresponding set of rook matrices by M ω ( S ). The set M | X | ( S ) is a Booleaninverse semigroup. Example 3.22.
Let = { , } , the two-element unital Boolean algebra. Then M n ( ) is the Boolean inverse monoid of n × n rook matrices in the sense of Solomon[17], and so is isomorphic to the finite symmetric inverse monoid on n letters I n .If G is a group, then G is the group with a zero adjoined. This is a Booleaninverse monoid we call a 0 -group. We proved in Proposition 3.2 that K fin ( G ) isa Boolean inverse semigroup for any discrete groupoid G . In the case where thegroupoid G is connected, we can say more. Proposition 3.23.
Let G be a connected groupoid. Then G = X × G × X , where G is a group and X is a non-empty set, and the Boolean inverse semigroup K fin ( G ) is isomorphic to M | X | ( G ) . INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 11
Proof.
We first apply the structure theorem for connected groupoids Lemma 2.1.We define an isomorphism θ : K fin ( G ) → M | X | ( G ) . Let A ∈ K fin ( G ). If A is the empty set then we map it to the | X | × | X | zero matrix.If A is non-empty then define θ ( A ) to be the | X | × | X | matrix with entries from G where ( x, g, y ) ∈ A if and only if the ( x, y )-element of θ ( A ) is g ; all other entriesare zero. Clearly, θ ( A ) has only a finite number of non-zero entries. Because A isa local bisection, each row of θ ( A ) contains at most one non-zero element; likewise,each column of θ ( A ) contains at most one non-zero element. This defines θ whichis clearly an injection. We prove that it is a bijection. Let A ′ ∈ M | X | ( G ). If it isthe zero matrix then we define A to be the empty set. Then we define A as follows:( x, g, y ) ∈ A precisely when the ( x, y )-element of θ ( A ) is g . Because A ′ is a rookmatrix, it follows that A is a finite local bisection. We have therefore proved that θ is a bijection. It remains to show that it is a homomorphism. Let A, B ∈ K fin ( G ).Then it is routine to check that θ ( AB ) = θ ( A ) θ ( B ). (cid:3) Notation • If S is an inverse semigroup, then E ( S ) denotes the semilattice of idempo-tents of S . If X ⊆ S then E ( X ) = E ( S ) ∩ X . • If S is an inverse semigroup, then A ( S ) denotes the set of atoms of S . • If X is a finite set, then I ( X ) is the symmetric inverse monoid on X . If X has n elements then the symmetric inverse monoid on X is also denoted by I n . These semigroups are finite Boolean inverse monoids. • If X is an infinite, set then I fin ( X ) denotes the Boolean inverse semigroupof all partial bijections of X with finite domains. • If S is an inverse semigroup then G ( S ) is called the restricted groupoid of S . It is equal to the set S with the restricted product if S does not have azero and the set S ∗ with respect to the restricted product if S does have azero. • K ( G ) is the Boolean inverse monoid of all local bisections of the finitediscrete groupoid G . • K fin ( G ) is the Boolean inverse semigroup of all finite local bisections of thediscrete groupoid G . • B ( S ) is the Booleanization of the inverse semigroup S . • G u ( S ) is Paterson’s universal groupoid of the inverse semigroup S . • G ( S ) is the ´etale groupoid of all prime filters of the Boolean inverse semi-group S . • KB ( G ) is the Boolean inverse semigroup of compact-open local bisectionsof the Boolean groupoid G .4. Finite Boolean inverse monoids
The structure of finite Boolean inverse monoids.
Finite Boolean in-verse semigroups are, of course, finite Boolean inverse monoids . The following is acorollary of Proposition 3.2 and will motivate the whole of this section; in fact, weshall prove that every finite Boolean inverse monoid is of this form. Observe thatthe natural partial order in K ( G ) is subset inclusion and so the atoms are the localbisections which are singleton sets. Proposition 4.1.
Let G be a finite (discrete) groupoid. Then K ( G ) , the set of alllocal bisections of G under subset multiplication, is a finite Boolean inverse monoid,the set of atoms of which forms a groupoid isomorphic to G . Let S be a finite Boolean inverse monoid. Our goal is to describe the structureof S in terms of its sets of atoms, just as in the case of finite Boolean algebras.Whereas in finite Boolean algebras the atoms simply form a set, in Boolean inversemonoids the set of atoms forms a groupoid. Denote by A ( S ) the set of atoms of S .The proof of the following is immediate by finiteness. Lemma 4.2.
Let S be a finite Boolean inverse monoid. Then each non-zero ele-ment is above an atom. The set of atoms is not merely just a set but in fact a groupoid.
Lemma 4.3.
Let S be an inverse semigroup with zero. Then the set of atoms, ifnon-empty, forms a groupoid under the restricted product.Proof. Let a, b ∈ A ( S ) such that d ( a ) = r ( b ). We prove that a · b is an atom. Let x ≤ a · b . Then x = a ′ · b ′ where a ′ ≤ a and b ′ ≤ b by Lemma 2.8. Now, a and b areatoms. Thus either x = a · b or x = 0. It follows that a · b is an atom as well. (cid:3) We now connect elements with the atoms beneath them. Let a ∈ S . Define θ ( a ) = a ↓ ∩ A ( S ). Lemma 4.4.
Let S be a Boolean inverse semigroup. For each a ∈ S , the set θ ( a ) is a local bisection of the groupoid A ( S ) .Proof. If a = 0 then θ ( a ) = ∅ . If a = 0 then it is above at least one atomby Lemma 4.2 and so is non-empty. Let x, y ∈ θ ( a ) such that d ( x ) = d ( y ). Since x, y ≤ a we deduce that a = b by Lemma 2.2. Dually, if r ( x ) = r ( y ) then x = y . (cid:3) If S is a finite Boolean inverse monoid, then we have defined a function θ : S → K ( A ( S )). Observe that θ (0) = ∅ . Our first main theorem is the following; it showsthat the finite Boolean inverse monoids described in Proposition 4.1 are typical. Theorem 4.5.
Let S be a finite Boolean inverse monoid. Then S is isomorphicto the Boolean inverse monoid K ( A ( S )) .Proof. By Lemma 4.2 and Lemma 4.3, the set A ( S ) of atoms of S forms a groupoid.By Lemma 4.4, the function θ is well-defined. We shall prove that θ is an isomor-phism. First, θ is a homomorphism. Let x be an atom such that x ≤ ab . Then x = a ′ · b ′ where a ′ ≤ a and b ′ ≤ b by Lemma 2.8. It follows that a ′ and b ′ areatoms by Lemma 2.3. We have therefore proved that θ ( ab ) ⊆ θ ( a ) θ ( b ). Conversely,let x ∈ θ ( a ) and y ∈ θ ( b ) such that the restricted product x · y is defined. Then x · y = xy ≤ ab by Lemma 2.8. It remains to prove that θ is a bijection. We showfirst that a = W θ ( a ). Put b = W θ ( a ). Then, clearly, b ≤ a . Suppose that b = a .Then a \ b = 0. Thus a \ b is above an atom x by Lemma 4.2. But then x ≤ a .It follows that x ≤ b which is a contradiction. Thus a = W θ ( a ). It follows that θ is an injection. Now let A ∈ K ( A ( S )). Then A is a set of pairwise orthogonalelements. Put a = W A . Clearly, A ⊆ θ ( a ). Let x be an atom such that x ≤ a .Then x = W a ∈ A ( x ∧ a ) by Lemma 3.3. It follows that x = a for some a ∈ A . Wehave therefore proved that θ ( a ) = A . (cid:3) Let G be a finite groupoid. We now relate the structure of the Boolean inversemonoid K ( G ) to the structure of the finite groupoid G . Lemma 4.6. (1)
A finite Boolean inverse monoid is fundamental if and only its groupoid ofatoms is principal. (2)
A finite Boolean inverse monoid is -simplifying if and only if its groupoidof atoms is connected. INITE AND SEMISIMPLE BOOLEAN INVERSE MONOIDS 13
Proof. (1) Let S be fundamental. Suppose that e a −→ e where e is an atomicidempotent. Then a is an atom by Lemma 2.3 Let f be any idempotent. Then f a ≤ a . It follows that f a = 0 or f a = a . Suppose that f a = 0. Then f e = 0and so af = 0. Thus f a = af . Suppose now that f a = a . Then f e = e = f e and so af = a . It follows again that f a = af . We have therefore proved that a commutes with every idempotent. and so, by assumption, a = e . Conversely,suppose that e a −→ e , where e is an atomic idempotent, implies that a = e . Let a commute with all idempotents. We prove that S is fundamental by showing that a is an idempotent. We can write a = W mi =1 a i where the a i are atoms and byProposition 3.8 we can assume this is an orthogonal join. We prove that d ( a i ) = r ( a i ) for all i from which the result follows. Since r ( a j ) a = a r ( a j ) we have that a j = W mi =1 a i r ( a j ). But a i r ( a j ) ≤ a j . Thus either a i r ( a j ) = 0 or a i r ( a j ) = a j .But a i r ( a j ) ≤ a i . It follows that a i r ( a j ) = a i and so a i = a j . Thus a j r ( a j ) = a j .Hence d ( a j ) ≤ r ( a j ) and so d ( a j ) = r ( a j ). By assumption a j is an idempotent. Itfollows that a is an idempotent.(2) Let S be a finite Boolean inverse monoid. Suppose first that it is 0-simplifyingand let e and f be any two atoms. Then, by assumption and Lemma 3.11, we havethat e ≡ f . In particular, f (cid:22) e . There is therefore a pencil from f to e . But,since f and e are both atoms, it follows that f D e . Thus all atomic idempotentsare D -related. Conversely, let S be a finite Boolean inverse monoid in which all itsatomic idempotents are D -related. Let e and f be any two non-zero idempotentsof S . Since S is finite, the idempotent e is the join of all the atoms that lie beneathit, as is f . Let the atoms beneath e be e , . . . , e n . Then each e i is D -related toan atom beneath f . But this shows that e (cid:22) f . By symmetry, we deduce that e ≡ f . (cid:3) The following combines [7, Theorem 4.18] and [14].
Theorem 4.7.
Let S be a finite Boolean inverse monoid. Then there are finitegroups G , . . . , G r and natural numbers n , . . . , n r such that S ∼ = M n ( G ) × . . . × M n r ( G r ) . Proof.
Let S be a finite Boolean inverse monoid with groupoid of atoms G . Then S ∼ = K ( G ) by Theorem 4.5. Let G , . . . , G r be the finite set of connected componentsof G . Then S ∼ = K ( G ) × . . . × K ( G r ) since the intersection of a local bisection witheach connected component G i is a local bisection of G i . Each groupoid G i isconnnected. The result now follows by Proposition 3.23. (cid:3) If we combine the above theorem with Lemma 4.6 and Example 3.22, we obtainthe following. Parts (2) and (3) were first proved in [7].
Corollary 4.8. (1)
The finite -simplifying Boolean inverse monoids are isomorphic to M n ( G ) where G is a finite group. (2) The finite fundamental Boolean inverse monoids are isomorphic to the finitedirect products I n × . . . × I n r . (3) The finite simple Boolean inverse monoids are isomorphic to I n . Booleanizations of finite inverse semigroups.
We shall now relate ar-bitrary finite inverse semigroups to finite Boolean inverse monoids. Let S be aninverse semigroup with zero. Then there is a universal homomorphism β : S → B ( S )to the category of Boolean inverse semigroups by [10]. The semigroup B ( S ) is calledthe Booleanization of S . If S does not have a zero then adjoin one to get S andthen the Booleanization of S is the same as the Booleanization of S . We now compute the Booleanization of a finite inverse semigroup. We shallneed the following notation. Let S be any inverse semigroup and let a ∈ S . Defineˆ a = a ↓ \ { a } ; this is just the set of all elements below a but excluding a Lemma 4.9.
Let S be a finite inverse semigroup and let α : S → T be a homomor-phism to a Boolean inverse semigroup T . Let a, b ∈ S be any elements such that d ( a ) = d ( b ) and r ( a ) = r ( b ) ; this means precisely that neither a − b nor ab − arerestricted products. Let ˆ a = { a , . . . , a m } and ˆ b = { b , . . . , b n } . Then the elements x = α ( a ) \ ( α ( a ) ∨ . . . ∨ α ( a m )) and y = α ( b ) \ ( α ( b ) ∨ . . . ∨ α ( b n )) are orthogonal.Proof. We shall prove that x − y = 0; the proof that xy − = 0 follows by symmetry.We have that x − y = α ( a − b ) \ ( α ( a − b ) ∨ . . . ∨ α ( a − b n ) ∨ α ( a − b ) ∨ . . . ∨ α ( a − m b )) . We know that a − b cannot be a restricted product. It follows by Lemma 2.8 thateither a − b = a − b j , for some j , or a − b = a − i b , for some i . In both cases, it isimmediate that x − y = 0. (cid:3) Lemma 4.10.