# Distributive inverse semigroups

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1 Distributive inverse semigroups Mark V Lawson Heriot-Watt University and the Maxwell Institute for Mathematical Sciences Edinburgh, Scotland, UK August 2012 This talk is based on work carried out in collaboration with Johannes Kellendonk, Ganna Kudryavtseva, Daniel Lenz, Stuart Margolis, and Ben Steinberg. 1

2 What is this talk about? This talk is about the relationship between inverse semigroups and topological groupoids. It was motivated in general by a growing body of work in which inverse semigroups are used to construct C -algebras (Exel, Paterson, Renault, Resende etc) 2

3 It was motivated in particular by work of Kellendonk on tiling semigroups and topological groupoids carried out in by work of Lenz, motivated by the above, carried out in 2002 though only published in by work of Birget on constructing the Thompson groups from polycyclic inverse monoids published in

4 A to Z 1. Definitions 2. Non-commutative Stone dualities 3. Examples 4. Constructing distributive inverse semigroups 5. More examples 6. Booleanization 7. Some more examples 8. Concluding remarks 4

5 1. Definitions A semigroup S is said to be inverse if for each s S there exists a unique s 1 S such that s = ss 1 s and s 1 = s 1 ss 1. An inverse semigroup S is equipped with two important relations. s t is defined if and only if s = te for some idempotent e. Despite appearances ambidextrous. Called the natural partial order. Compatible with multiplication. s t if and only if st 1 and s 1 t both idempotents. Called the compatibility relation. It controls when pairs of elements are eligible to have a join. 5

6 Lattices need not have 1 s but always have 0 s. If they have 1 s they will be called unital. Thus: distributive lattices vs. unital distributive lattices; Boolean algebras vs. unital Boolean algebras. 6

7 A distributive inverse semigroup is one which has joins of compatible pairs of elements and multiplication distributes over such joins. A Boolean inverse semigroup is a distributive inverse semigroup with a Boolean algebra of idempotents. A Boolean inverse -semigroup is a Boolean inverse semigroup with the additional property that all pairs of elements have a meet. 7

8 A vanilla distributization. Theorem [Schein] Let S be an inverse semigroup. There is a distributive inverse semigroup D(S) and a map δ:s D(S) which is universal for maps from S to distributive inverse semigroups. 8

9 Let P be a poset with zero 0. A subset F P is a filter if it is downwardly directed and upwardly closed. It is proper if 0 / F; all filters will be proper. An ultrafilter is a maximal proper filter. A filter F is prime if a b F implies that a F or b F. 9

10 A groupoid G is a (for us, small) category with every arrow invertible. The set of identities (or objects) of G is denoted by G o. The o stands for objects. If a groupoid G carries a topology making the multiplication and inversion continuous, it is called a topological groupoid. The most important class of topological groupoids are the étale groupoids. We use Resende s characterization to define them. A topological groupoid G is étale if G o is an open set and the product of any two open sets in G is an open set. N.B. Hausdorffness is not assumed. 10

11 A topological space is said to be sober if each point of the space is uniquely determined by the open sets that contain it (plus a bit more.) A topological space X is said to be spectral if it is sober and has a basis of compact-open sets that is closed under finite non-empty intersections. We do not assume that X is compact. 11

12 An étale groupoid is called spectral if its space of identities is a spectral space. A étale groupoid is called Boolean if its space of identities is Boolean. 12

13 To avoid piling on definitions, morphisms will be kept in the background throughout this talk they can be defined so that things work. 13

14 2. Non-commutative Stone dualities Classical theorems. Theorem [Stone duality for distributive lattices] The category of distributive lattices and their proper homomorphisms is dually equivalent to the category of spectral spaces and their coherent continuous maps. A Hausdorff spectral space is called a Boolean space. Theorem [Stone duality for Boolean algebras] The category of Boolean algebras and their proper homomorphisms is dually equivalent to the category of Boolean spaces and their coherent continuous maps. 14

15 The staring point of our work. Theorem [Stone duality for distributive inverse semigroups] The category of distributive inverse semigroups is dually equivalent to the category of spectral groupoids. Theorem [Stone duality for Boolean inverse semigroups] The category of Boolean inverse semigroups is dually equivalent to the category of Boolean groupoids. Theorem [Stone duality for Boolean inverse - semigroups] The category of Boolean inverse -semigroups is dually equivalent to the category of Hausdorff Boolean groupoids. 15

16 Proof sketch Let G be a spectral groupoid. A local bisection A of a groupoid G is a subset such that A 1 A,AA 1 G o. The set of all compact-open local bisections is a distributive inverse semigroup. Let S be a distributive inverse semigroup. Let P be a prime filter. Define d(p) = (P 1 P) and r(p) = (PP 1 ). Define the partial product P Q to be (PQ) iff d(p) = r(p). In this way, the set of prime filters becomes a groupoid G P (S). Let s S. Define X s to be the set of all prime filters that contain s. These sets form the basis of a topology on G P (S). 16

17 Higher level proof A pseudogroup is an inverse semigroup with arbitrary non-empty compatible joins and infinite distributivity. There is an adjunction between the dual of the category of pseudogroups and the category of étale groupoids due to Resende (without morphisms) and Lawson/Lenz (with morphisms). The duality for distributive inverse semigroups results from this duality by restricting using coherence. We are therefore in the world of non-commutative frame theory. 17

18 3. Examples Let G be a finite discrete groupoid. The set of all local bisections of G is a finite Boolean inverse -semigroup I(G) and all finite inverse -semigroups are of this form. Write G = m i=1 G i where the G i are the connected components of G. Then I(G) = I(G 1 )... I(G m ). Let G be a finite connected discrete combinatorial groupoid and put G o = X. Then I(G) = I(X), a finite symmetric inverse monoid. The fundamental finite Boolean inverse -semigroups are therefore of the form Call these semisimple. I(X 1 )... I(X m ). May construct AF inverse monoids from Bratteli diagrams and injective morphisms between semisimple inverse monoids. 18

19 4. Constructing distributive inverse semigroups Let S be an inverse semigroup. Let a S and b 1,...,b m a. We say that the set of elements {b 1,...,b m } is a (tight) cover of a if for each 0 x a there exists b i such that 0 z x,b i for some z. A tight filter is a filter A such that if a A and {b 1,...,b m } covers a then b i A for some i. A semigroup homomorphism θ:s T to a distributive inverse semigroup is said to be a tight map if for each element a S and tight cover {a 1,...,a n } of a we have that θ(a) = ni=1 θ(a i ). 19

20 Intuitive idea The idea is to present distributive inverse semigroups by means of generators and relations. The generating set is in fact an inverse semigroup S. The relations are given by the tight covers if {b 1,...,b m } is a (tight) cover of a, then THINK a = m i=1 b i. 20

21 Theorem [Tight completions] Let S be an inverse semigroup. 1. There is a distributive inverse semigroup D t (S) and a tight map δ:s D t (S) which is universal for tight maps from S to distributive inverse semigroups. 2. There is an order isomorphism between the poset of tight filters in S and the poset of prime filters in D t (S) under which ultrafilters correspond to ultrafilters. We call the distributive inverse semigroup D t (S) the tight completion of S. 21

22 If the tight completion of an inverse semigroup is actually Boolean we say that the semigroup is pre-boolean. It can be proved that every ultrafilter is a tight filters. Theorem An inverse semigroup is pre-boolean if and only if every tight filter is an ultrafilter. 22

23 5. More examples The polycyclic monoid P n, where n 2, is defined as a monoid with zero generated by the variables a 1,...,a n,a 1 1,...,a 1 n subject to the relations a 1 i a i = 1anda 1 i a j = 0,i j. Every non-zero element of P n is of the form yx 1 where x and y are elements of the free monoid on {a 1,...,a n }. The product of two elements yx 1 and vu 1 is zero unless x and v are prefix-comparable in which case yx 1 vu 1 = { yzu 1 if v = xz for some z y(uz) 1 if x = vz for some z 23

24 The polycyclic monoid P n is a pre-boolean inverse monoid. The set {a 1 a 1 1,...,a na 1 n } is a tight cover of the identity, and in some sense, determines all other tight covers. Theorem The Boolean completion of P n is called (here) the Cuntz inverse monoid CI n. 1. This monoid is congruence-free. 2. Its group of units is the Thompson group V n,1. 3. Its associated groupoid is the groupoid also associated with the Cuntz C -algebra C n. 24

25 O. Bratteli, P. E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Memoirs of the A.M.S. No. 663, (1999) is, in fact, a study of tight maps from P n to I(X). 25

26 All Thompson-Higman groups V n,r can be constructed in a similar way. Self-similar groups actions give rise to generalizations of the polycyclic inverse monoids which are also pre-boolean. Finite directed graphs can be used to construct pre-boolean inverse semigroups. AF inverse monoids are generated by pre- Boolean inverse monoids. 26

27 6. Booleanization Let S be a distributive inverse semigroup and G P (S) its associated spectral groupoid. Recall that a basis is given by π = {X s :s S} where X s is the set of all prime filters containing s. Define Π = {X s Xt c :s,t S, t s}. It is convenient to define X s;t = X s Xt c where t s. The topology it generates is called the patch topology. We denote by G P (S) the groupoid G P (S) equipped with the patch topology. Theorem [Booleanization] G P (S) is a Boolean groupoid. 27

28 7. Yet more examples Paterson s universal groupoid Let S be an arbitrary inverese semigroup. Construct its vanilla distributization D(S). Form the spectral groupoid G P (D(S)). Booleanize G P (D(S)). This is Paterson s universal groupoid. Exel s tight groupoid Let S be an arbitrary inverese semigroup. Construct its tight completion D t (S). Form the spectral groupoid G P (D t (S)). Booleanize G P (D t (S)). This is Exel s tight groupoid. 28

29 8. Concluding remarks 1. It is natural to compute K 0 -groups of Boolean inverse monoids (ongoing with AW). This may throw light on the information contained in inverse semigroups of partial symmetries. 2. Our theory can be used to construct interesting groups of the Thompson-Higman variety. The elements of the group are obtained by glueing together partial bijections. 29

30 3. Finite semigroup theory benefited greatly from its associations with the theory of regular languages and their finite state automata. I propose that inverse semigroup theory be viewed from the perspective of non-commutative frame theory. This also provides natural connections with topos theory (Funk) and quantale theory (Resende) as well as with C -algebras. 30

31 Is that it? said Eeyore, Yes, said Christopher Robin. Is that what we were looking for? Yes, said Pooh. Oh! said Eeyore. Well, anyhow it didn t rain, he said. 31

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