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On Princesses and Decompositions Some Aspects of Synchronization Theory Artur Schfer University o St Andrews 17th Feb, 2015 Artur Schfer ( University o St Andrews ) On Princesses and Decompositions 1 / 14 17th Feb, 2015 Synchronization
Some Aspects of Synchronization Theory Artur Schäfer
University o St Andrews
17th Feb, 2015
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 1 / 14
1 2 3 4
RED BLUE RED BLUE RED BLUE RED
Follow BLUE, RED, BLUE, BLUE.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 2 / 14
Definition
A semigroup S is synchronizing, if it contains a constant map. (S a finite transformation semigroup acting on a finite set X) How does a constant map look like? c = g1a1g2a2g3a3...gkakgk+1 = ag1
1 ag1g2 2
ag1g2g3
3
...ag1g2...gk
k
g1g2g3...gkgk+1. (for ag = gag−1).
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 3 / 14
For S = G, a, for some group G and a transformation a, this reduces to c = ag1ag1g2...ag1g2...gkg1g2g3...gkgk+1 c is constant if and only if c′ is constant. c′ = ag1ag1g2...ag1g2...gk. Hence, S is synchronizing, if and only if aG ⊆ S is synchronizing.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 4 / 14
Definition
A group G normalizes a transformation a (G is a-normalizing), if G, a = aG.
Theorem: [Araujo, Cameron, Mitchell, Neunhöffer]
A group G normalizes all transformations a ∈ Tn \ Sn if and only if G is
1 {1}, An, Sn, or 2 one of five other groups. Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 5 / 14
Now, consider S = G, a1, a2, ..., an.
Definition
1 A group G is (a1, ..., an)-normalizing, if G, ai = aG
i , for all i.
2 A group G is {a1, ..., an}-normalizing, if G, a1, ..., an = aG
1 , ..., aG n .
3 A group G is strongly {a1, ..., an}-normalizing, if
G, aj1, ..., ajk = aG
j1, ..., aG jk, for any subset {aj1, ..., ajk}.
Properties: 3) ⇒ 2) and 1), but what about other relations.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 6 / 14
Assume we can decompose S as follows G, a1, ..., an \ G = G, a1 \ G ⊎ · · · ⊎ G, an \ G
Lemma
Given such a decomposition. Then, G is (a1, ..., ar)-normalizing implies G is {a1, ...ar}-normalizing.
Definition
T = {a1, ..., ar}, r ≥ 2.
1 G is T-decomposing if the above decomposition holds. 2 G is strongly T-decomposing if for all subsets T ′ of T, G is
T ′-decomposing.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 7 / 14
Again, T = {a1, ..., ar}, S = G, T \ G and S1 = G, a1 \ G. Then S − S1 is also a semigroup. -> “Differences” and “Sums” of semigroups are semigroups For semigroups which are strongly decomposable (a1, ..., ar)-normalizing and strongly {a1, ..., ar}-normalizing is the same. Hence, this gives a close relation to semigroups of the form aG
1 , aG 2 , ..., aG r .
Most importantly: The effort of analysing G, a1, ..., ar is reduced to an analysis of G, ai, for all i.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 8 / 14
By construction, in the framework of this strong decomposition holds: (a1, ..., ar)-normalizing ⇔ strongly {a1, ..., ar}-normalizing . And both imply {a1, ..., ar}-normalizing . Simplified properties:
1 check “regularity”; 2 check “idempotent generated”; 3 check minimal generating sets; 4 check automorphism groups (for some cases).
To Do:
1 Derive L−, R−, D−classes of S from its compontents; 2 Analysis of subsemigroups 3 ... Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 9 / 14
Lemma
If a semigroup is decomposable (like above), then it is not synchronizing. The best known examples of non-synchronizing semigroups come from endomorphism monoids of graphs.
Theorem [Cameron]
G, a is non-synchronizing, if and only if it contains endomorphisms of some graph (non-trivial with complete core).
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 10 / 14
Hamming Graphs: The endomorphism monoid of a Hamming graph consists of either bijections or Latin squares and Latin hypercubes. 1st case: 2 dimensions → the endomorphisms are Latin squares. S′ = S \ G is a simple semigroup. S′ is strongly decomposable → we can find easily a minimal generating set T (S′ = G, T \ G). If S′ = S1 ⊎ S2, then for s1 ∈ S1 and s2 ∈ S2 holds s1s2 ∈ S1 and s2s1 ∈ S2. → ∞-family of examples for the strong decomposition
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 11 / 14
2nd case: m dimensions consider the minimal ideal. → same as in dimension 2, but with Latin hypercubes. 3rd case: m dimensions, consider the whole endomorphism monoid (non-bijective part) S′. → We can always find a (non-strong) decomposition.
Example
S′ = Sing(H(3, 4)); S′ contains transformations of ranks 4 and 16 only. -> We can find a decomposition into 5 parts which is not strong. S′ = S1 ⊎ S2 ⊎ S3 ⊎ S4 ⊎ S5. Here, surprisingly holds: s1s2 ∈ S3, for some s1 ∈ S1 and s2 ∈ S2.
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 12 / 14
Orthogonal array graphs (Latin square graphs). Their endomorphisms are Latin squares. The triangular graph is the graph where the vertices are the subsets
Its endomorphisms are Latin squares. We determined all (primitive) graphs ≤ 64 vertices which have proper endomorphisms (106 graphs). And we know the monoids of the smallest graphs (all up to 49 vertices). →All of the have the decomposition property (or are simply generated).
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 13 / 14
Artur Schäfer (University o St Andrews) On Princesses and Decompositions 17th Feb, 2015 14 / 14