representing inverse semigroups by block permutations
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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