Representing inverse semigroups by block permutations

1. Representing inverse semigroups by block permutations

2. What are they? 4 ways to imagine: - bijections between quotient sets of X; or - “chips”; or - diagrams; or - relations, bifunctional and full.

3. Example of a block perm.

4. Some properties • these form i.s., I*X - with the ‘right’ mult’n. • every i.s. embeds in some I*X • interesting

5. 4 inspirations and spurs for project: • (1) B M Schein’s theory: reps of S in IX • (2) B M Schein’s challenge: describe/classify all (trans. eff.) reps of S by relations • (3) advocacy for the dual, I*X • (4) efficiency of a representation -- how many points?

6. Inspiration: • (1) B M Schein’s theory: reps of S in IX - all are sums of transitive effective ones; these are obtained from action on cosets • (2) B M Schein’s challenge: describe/classify all (trans. eff.) reps of S by binary relations - using composition as multiplication

7. (3) advocacy for the dual, I*X - SIM in the cat. Setopp ; - test-bed for other ‘natural’ contexts for reps., e.g. partial linear

8. (4) efficiency of a representation -- how many points? f : S  IX , degf = card X degS = min {degf : f faithful } f : S  I*X , deg*f = card X deg*S = min {deg*f : f faithful } Now there exist faithful reps ...

9. f : In I*n+1 , (the extra point is a sink for all pts ‘unused’ in a partial bij.) f : I*n IN , whereN = 2n - 1 - 1 (V.Maltcev; Schein again!) Both best possible. So deg*S ≤ degS + 1 always, while there are some S such that degS >> deg*S .

10. Rephrase B M Schein’s challenge? : describe all transitive effective reps of S in I*X But what do transitive and effective mean in I*X? Let S be an inverse subsgp of I*X (to simplify)

11. Imitating the classical case: Say S is (weakly) effective if S is not contained in any proper local monoid I*X of I*X . Note: I*X  I*X/

12. Let P = set of primitive ips. in I*X e.g.  = ( 1 | 2, 3, 4 ) Define the transitivity reln on P TS = {(p, q) : s-1ps = q for some s in S }; only a partial equivalence. [y, ps = sq  0 ] Say S is (weakly) transitive if TS is total on its domain. [ Classical case: total on P ]

13. Let TS-classes be Piand define iby si = { ps : p in Pi}. Si ≤ a local monoid, and s = isi , all s. (So S ≈ ‘product’ of Si ) However, iis only a pre-homomorphism [ = lax hom., i.e., (st)i ≤si ti ]

14. Take p, q inPi . So there is s such that psq ≠ 0. Then p(si)q ≠ 0 --- so Si is transitive in the weak sense.

15. Seeking internal description of transitives For A  S, [A] = {x : x  a, some a in A} • Coset: [Ha] with aa-1 in [H] • Let X be the set of all cosets

16. A rep. : • s= {( [Ha], [Hb] ) : [Has] = [Hbs-1s]} --- where [Ha], [Hb] are cosets •   s tos is a rep. of S in I*X

17. An example : S =

18. Ex., ctd  = ( 1 | 2 3 4 )

19. (2 | 134) annihilates all, i.e. is in domain of no element of Sbut S fits in neither relevant local monoid so only weakly effective

20. The two orbits are P1 = { (12 | 34), (13 | 24) } ; and P2 = {(1 | 234)}. The local monoids they generate are not 0-disjoint!

21. The maps i : 1 fixes    and maps  to zero, 2 fixes  and maps all of   to zero. True homs in this case. Why??

22. [Subsgps] and their cosets

23. The maps 

24. Details in a draft discussion paper on the UTas e-print site