Representing inverse semigroups by block permutations.

Representing inverse semigroups by block

permutations

What are they?

4 ways to imagine:

- bijections between quotient sets of X; or

- “chips”; or

- diagrams; or

- relations, bifunctional and full.

Example of a

block perm.

Some properties

• these form i.s., I*X - with the ‘right’ mult’n.

• every i.s. embeds in some I*X

• interesting

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?

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

• (3) advocacy for the dual, I*X

- SIM in the cat. Setopp ;

- test-bed for other ‘natural’ contexts for reps., e.g. partial linear

(4) efficiency of a representation -- how many points?

f : S IX , deg f = card X

deg S = min {deg f : f faithful }

f : S I*X , deg* f = card X

deg*S = min {deg* f : f faithful }

Now there exist faithful reps ...

f : In I*n+1 , (the extra point is a sink for all pts ‘unused’ in a partial bij.)

f : I*n IN , where N = 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 .

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)

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/

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 ]

Let TS-classes be Pi and define i by

si = { ps : p in Pi }.

Si ≤ a local monoid, and s = i si , all s.

(So S ≈ ‘product’ of Si )

However, i is only a pre-homomorphism

[ = lax hom., i.e., (st)i ≤ si ti ]

Take p, q in Pi .

So there is s such that psq ≠ 0.

Then p(si)q ≠ 0

--- so Si is transitive in the weak sense.

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

A rep. :s =

{( [Ha], [Hb] ) : [Has] = [Hbs-1s]}

--- where [Ha], [Hb] are cosets

• s to s is a rep. of S in I*X

An example : S =

Ex., ctd

=

( 1 | 2 3 4 )

(2 | 134) annihilates all, i.e. is in domain of no element of S but

S fits in neither relevant local monoid

so only weakly effective

The two orbits are

P1 = { (12 | 34), (13 | 24) } ; and

P2 = {(1 | 234)}.

The local monoids they generate are not 0-disjoint!

The maps i :

1 fixes and maps to zero,

2 fixes and maps all of to zero.

True homs in this case. Why??

[Subsgps] and their cosets

H cosets

[ ] = S S

[ ] = { } { }

[ -1 ] = { -1 } { -1 }, { }

[-1 ] = {-1 } {-1 }, { }

The maps

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