Partial Transformations: Semigroups, Categories and Equations. Ernie Manes, UMass, Amherst Semigroups/Categories workshop U Ottawa, May 2010

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1 1 Partial Transformations: Semigroups, Categories and Equations Ernie Manes, UMass, Amherst Semigroups/Categories workshop U Ottawa, May 2010

2 2 What is a zero? 0 S, 0x =0=x0 0 XY : X Y, W f X 0 XY Y g Z = 0 WZ How is total defined? Given the zero, an element or morphism f is total if fg =0 g =0 monic (= left cancellable) total. g,f total gf total f total.

3 3 In PT(X) what is the domain of f : X X? fx = x if fx is defined and fx is otherwise undefined. How do we get f from the semigroup structure of PT(X)? Work of Fountain, Gomes, Gould, Jackson, Pastijn, Schweizer, Sklar, Stokes and many others from 1960 on: If f has a unique inverse, f = f 1 f. E = {f : f has a unique inverse} is a subsemilattice. For any f, f is the unique element of E with f g, if x y means e Exe= x ye = y.

4 4 In the category PT, what is the domain of f : X Y? Theorem For f : X Y, coproduct X = D f + K f with D f i X j K f total f 0 Y and D f, K f are unique as summands. Then f is defined by the coproduct property by D f i X j K f i f 0 X Note: K f = eq(f,0). So, since summands form a Boolean algebra, D f = eq(f,0).

5 The semigroup approach led to the study of new classes of semigroups such as (left/right/both) 5 principally projective (semi)-adequate (weakly) E-ample type A abundant restriction semigroup

6 The category approach generalizes to restriction categories. The particular construction for f above is available for the partial map category of any Boolean topos (Cockett and Manes 2009). 6 An endomorphism monoid in such a category provides potentially new examples of semigroups. One completes a semigroup by representing it as an endomorphism monoid in a category and using category-theoretic tools!

7 7 Restriction categories (Cockett and Lack, 2002). X f Y X f X (R.1) f f = f. (R.2) Y f X g Z, f g = g f. (R.3) X (R.4) X f Y f,g Y, g f = g f. g Z, gf = f gf. See Cockett and Manes for (embarrassing) historical remarks. This is a varietal extension of categories.

8 8 Restriction categories are ordered categories via f g if g f = f f is total if f = id. If 0 exists with 0 = 0 then, if f is total in this sense and if fz = 0, then. z = id z = fz= z fz = z 0=0 The converse is true with more assumptions (later). Total maps are maximal.

9 9 Monoids are one-object categories. Semigroups are monoids, namely those with no nontrivial invertibles. Note that concepts defined by universal mapping properties in a semigroup without unit are not just unique up to isomorphism, they are unique! Hence S is a restriction semigroup if S 1 is a one-object restriction category.

10 10 Weakly Right E-Ample Semigroups (Victoria Gould web site notes, 2002) As before, in a semigroup S with given set E of idempotents, write x y if for all e E, xe = x ye = y. Axioms: For all x there exists unique e E with x e, call it x. x y xz yz. yx= x yx. Theorem If (S, E) is weakly right E-ample, x makes S a restriction semigroup. If S is a restriction semigroup, E = {x : x = x}, S is weakly right E-ample. The two ideas are the same!

11 11 Embedding a Restriction Semigroup in PT(X): Inverse semigroup inverse-closed in I(X) (Vagner- Prestion). Right ample semigroup f-closed in I(X). Weakly right ample semigroup f-closed in PT(X) such that every idempotent has form f. Restriction semigroup f-closed in PT(X). (Jackson and Stokes). The last one shows that the variety generated by all PT(X) is precisely restriction semigroups. So, for example, there is no way to add equations to (R.1,...,R.4) to get regular restriction semigroups.

12 What Properties of Inverse Semigroups Generalize? 12 For inverse semigroups, say x yif xy 1 is idempotent. This is equivalent to x y = y x which makes sense for all restriction semigroups. We say f,g are compatible. A restriction semigroup is E-unitary if the set of restriction idempotents is an upper set. For an inverse semigroup, E-unitary is transitive. For a restriction semigroup, is easy and the converse is currently unsettled.

13 An inverse semigroup with one idempotent is a group. A restriction semigroup with one restriction idempotent is a monoid with x =1. 13 In both cases, there is a largest congruence σ on S such that S\σ has one idempotent. it is xσy e = e with xe = ye Moreover, in both cases, σ is the transitive closure of. Such σ for all idempotents gives the left cancellative monoid reflection of a Type A semigroup (Fountain 1977). Open Question: For inverse semigroups, define xµy e = e, x 1 ex = y 1 ey Then µ is the largest idempotent-separating congruence. Is there such a µ for restriction semigroups? We do, however, have

14 14 The McAlister P -Theorem Theorem (McAlister 1974) If S is an inverse semigroup, there exists an E-unitary inverse semigroup P, finite if S is, and a surjective homomorphism P S which separates idempotents. Theorem (Manes 2006) If S is a restriction semigroup, there exists an E-unitary restriction semigroup P, finite if S is, and a surjective homomorphism P S which separates restriction idempotents. In both cases, C is proper and proper implies E-unitary. The definition of proper is that if some restriction idempotent e exists with xe = ye, then x = y.

15 15 Splitting Idempotents The idempotent completion of a restriction category (for all restriction idempotents) is a restriction category with the restriction of t : e f defined to be e t : e e. The idempotent completion of a restriction semigroup is a subcategory of the category of sets. In an inverse monoid, the only idempotent that splits is 1. Observation Split restriction categories are a varietal extension of category theory.

16 16 What can be say about interesting operations and equations on PT(X, Y )? A Locally Boolean semilattice (LBS) is a meet semilattice X with 0 such that x X [0,x]= x is a Boolean algebra. An LBS with a greatest element is the same thing as a Boolean algebra. An LBS in which each two elements have an upper bound is the same thing as a Boolean ring. If f,g u the supremum f g in [0,u] is the supremum in X. PT(X, Y ) is a natural example of an LBS which is not a Boolean ring. We have f g = fx (or gx) for x such that fx,gx are both defined and equal. Note: the domain of f g is not eq(f,g). PT(X, Y ) (2 X Y, ) is an ideal (= downset). 0 is the empty function [0,f] = 2 D f Any downset in an LBS is an LBS.

17 17 LBS is a variety Binary x y (thought of as the relative complement x\(x y) in [0,x]). Nullary 0 (but note: x x = 0). x y is an abbreviation for x (x y). The -preserving maps f preserve 0, and induce Boolean algebra homomorphisms [0,x] [0,fx] because, for y x, fy = f(x y) =fx fy =(fy).

18 18 Equations defining the variety x y abbreviates x (x y). < u, x, y > abbreviates u [(u x) (u y)]. x y = y x (x y) z = x (y z) x x = 0 x (u x) = 0 x (u x) = x x (x y) = x y (u x) (u (x y)) = u x < u, x, y z> = < u, x, y > < u, x, z > Other interesting equations include (x z) (y z) = (x y) z (x z) (y z) = (x y) z

19 Theorem Every LBS L is a sub-lbs of a Boolean algebra. Proof: For a L, L ψ a [0,a], x a x is an LBS-morphism. is the desired embeddiing. L [ψ a] a [0,a] 19 Boolean rings is a full subcategory of all LBS, so this theorem also embeds a Boolean ring in a Boolean algebra. Corollary 2 is a cogenerator in LBS. Proof: Embed the LBS in a Boolean algebra and embed the Boolean algebra in a power of 2. Corollary Any functor LBS X which preserves products and equalizers has a left adjoint. Proof. Special adjoint functor theorem.

20 20 The following varietal extension of category theory is a fuller monty for partial transformations. First notice that the restriction idempotents X X is [0,id X ]. We have the abbreviations = id f f g = id ((id f) (id g)) f Definition An LBS-restriction category is a restriction category with each X(X, Y ) an LBS subject to the axioms (FM.1) For W h X f,g Y,(f g)h = fh gh. (FM.2) For X f,g Y h Z, h(f g) = hf hg. (FM.3) f g = f f g. (FM.4) For W h X f Y,(fh) = h f h. Example PT, PT(X). Example The partial morphism category of any Boolean topos. A one-object LBS-restriction category is an LBS-monoid.

21 21 Some properties of an LBS-restriction category The least elements 0 XY : X Y form zero maps which are a restriction zero, that is, 0 XY =0 XX. Earlier: f = id X (fz =0 z = 0). Now the converse: As f f =0, f =0sof = id X.

22 22 What can be said about joins? In PT(X), joins don t always exist. For example, distinct total functions can t have an upper bound. f g exists f g; the latter means f g = g f. A special case of f gis f g which means f g =0. Objective Find a varietal extension of restriction categories which includes the partial morphism category of a Boolean topos as an example and which satisfies the following for W k X f,g Y h Z with f g: f g exists. k(f g) =kf kg. (f g)h = fh gh. Note: f g kf kg and fh gh in any restriction category.

23 23 Here s a Solution Adjoin to split LBS-restriction categories a new binary f g : X Y for f,g : X Y subject to (.1) (f g)h = fh gh. (.2) f g = f g. (.3) (u f) (u g) u. (.4) f f g. Theorem PT is such a category if f g is f g f. In general, f g exists f gin which case f g = f g = f g f. To prove f g f exists one uses splitting of the idempotents f, f.

24 24 Open Question on Completeness Theorem (Jackson and Stokes) If S is a restriction semigroup, a S, define λ a PT(S) with domain {t : at = t} by λ a t = at. Then S ψ PT(S), is a restriction embedding. a λ a Unfortunately, if S LBS, ψ preserves but need not preserve 0or. Can an LBS be LBS-embedded in a PT(X)? If not, is the subvariety of LBS generated by the various PT(X) all of LBS? etc.

25 Open Question What is the general theory of varietal extensions of categories? 25 That s it! Thank you for listening/not listening quietly.