On Princesses and Decompositions Some Aspects of Synchronization Theory Artur Schäfer University o St Andrews 17th Feb, 2015 Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 1 / 14 17th Feb, 2015
Synchronization 1 RED BLUE 2 4 RED BLUE RED BLUE RED 3 Follow BLUE, RED, BLUE, BLUE. Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 2 / 14 17th Feb, 2015
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 = g 1 a 1 g 2 a 2 g 3 a 3 ... g k a k g k + 1 = a g 1 1 a g 1 g 2 a g 1 g 2 g 3 ... a g 1 g 2 ... g k g 1 g 2 g 3 ... g k g k + 1 . 2 3 k (for a g = gag − 1 ). Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 3 / 14 17th Feb, 2015
For S = � G , a � , for some group G and a transformation a, this reduces to c = a g 1 a g 1 g 2 ... a g 1 g 2 ... g k g 1 g 2 g 3 ... g k g k + 1 c is constant if and only if c ′ is constant. c ′ = a g 1 a g 1 g 2 ... a g 1 g 2 ... g k . Hence, S is synchronizing, if and only if � a G � ⊆ S is synchronizing. Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 4 / 14 17th Feb, 2015
Normalizing Groups Definition A group G normalizes a transformation a ( G is a -normalizing), if � G , a � = � a G � . Theorem: [Araujo, Cameron, Mitchell, Neunhöffer] A group G normalizes all transformations a ∈ T n \ S n if and only if G is one of 1 { 1 } , A n , S n , or 2 one of five other groups. Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 5 / 14 17th Feb, 2015
Now, consider S = � G , a 1 , a 2 , ..., a n � . Definition 1 A group G is ( a 1 , ..., a n ) - normalizing , if � G , a i � = � a G i � , for all i . 2 A group G is { a 1 , ..., a n } -normalizing, if � G , a 1 , ..., a n � = � a G 1 , ..., a G n � . 3 A group G is strongly { a 1 , ..., a n } -normalizing, if � G , a j 1 , ..., a j k � = � a G j 1 , ..., a G j k � , for any subset { a j 1 , ..., a j k } . Properties: 3 ) ⇒ 2 ) and 1 ) , but what about other relations. Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 6 / 14 17th Feb, 2015
Disjoint Decompositions Assume we can decompose S as follows � G , a 1 , ..., a n � \ G = � G , a 1 � \ G ⊎ · · · ⊎ � G , a n � \ G Lemma Given such a decomposition. Then, G is ( a 1 , ..., a r ) -normalizing implies G is { a 1 , ... a r } -normalizing. Definition T = { a 1 , ..., a r } , 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 7 / 14 17th Feb, 2015
Strong Decomposition Again, T = { a 1 , ..., a r } , S = � G , T � \ G and S 1 = � G , a 1 � \ G . Then S − S 1 is also a semigroup. -> “Differences” and “Sums” of semigroups are semigroups For semigroups which are strongly decomposable ( a 1 , ..., a r ) -normalizing and strongly { a 1 , ..., a r } -normalizing is the same. Hence, this gives a close relation to semigroups of the form � a G 1 , a G 2 , ..., a G r � . Most importantly: The effort of analysing � G , a 1 , ..., a r � is reduced to an analysis of � G , a i � , for all i . Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 8 / 14 17th Feb, 2015
So far ... By construction, in the framework of this strong decomposition holds: ( a 1 , ..., a r ) -normalizing ⇔ strongly { a 1 , ..., a r } -normalizing . And both imply { a 1 , ..., a r } -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 9 / 14 17th Feb, 2015
Which semigroups are (strongly) decomposable? And what is the difference? 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 10 / 14 17th Feb, 2015
Examples 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 ′ = S 1 ⊎ S 2 , then for s 1 ∈ S 1 and s 2 ∈ S 2 holds s 1 s 2 ∈ S 1 and s 2 s 1 ∈ S 2 . → ∞ -family of examples for the strong decomposition Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 11 / 14 17th Feb, 2015
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 ′ = S 1 ⊎ S 2 ⊎ S 3 ⊎ S 4 ⊎ S 5 . Here, surprisingly holds: s 1 s 2 ∈ S 3 , for some s 1 ∈ S 1 and s 2 ∈ S 2 . Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 12 / 14 17th Feb, 2015
More Examples Orthogonal array graphs (Latin square graphs). Their endomorphisms are Latin squares. The triangular graph is the graph where the vertices are the subsets of size two of { 1 , ..., n } . Two vertices are adjacent if the sets intersect. 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 13 / 14 17th Feb, 2015
Thank You for Your Attention Artur Schäfer ( University o St Andrews ) On Princesses and Decompositions 14 / 14 17th Feb, 2015
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