Maximal subsemigroups via independent sets Wilf Wilson 26 th April - PowerPoint PPT Presentation

Maximal subsemigroups via independent sets Wilf Wilson 26 th April 2017

Maximal subsemigroups? 1 2 3 A maximal subsemigroup is * * * formed by removing parts of one D -class. 5 7 It has one of several forms. * * However: A semigroup acts on itself. 6 Elements can generate parts * of lower D -classes. These things limit the maximal subsemigroups that occur. 4 * * * * * * * * *

My contributions C. Donoven, J. D. Mitchell, and W. A. Wilson Computing maximal subsemigroups of a finite semigroup arXiv:1606.05583 J. East, J. Kumar, J. D. Mitchell, and W. A. Wilson Maximal subsemigroup of finite transformation and partition monoids In preparation

The general technique Focus on some ‘nice’ monoids! To find the maximal subsemigroups from a D -class: Construct a graph that captures the action on L -/ R -classes. Compute the maximal independent subsets. Find the vertices that are not adjacent to a vertex of degree 1 .

Partial transformations Reminders: A partial transformation of degree n is a partial map on { 1 , . . . , n } . A partial transformation has a domain , a kernel , and an image . A total transformation has domain { 1 , . . . , n } . Order-preserving: i ≤ j implies ( i ) f ≤ ( j ) f . Order-reversing: i ≤ j implies ( i ) f ≥ ( j ) f . Notation for Green’s classes of rank n − 1 : L i L -class of elements with image { 1 , . . . , n } \ { i } . R i R -class of elements with domain { 1 , . . . , n } \ { i } . R { i,j } R -class of elements with non-trivial kernel class { i, j } .

Order-preserving partial transformations n � n �� n + k − 1 � � |PO n | = k k k =0 { R { 1 , 2 } } { R { 2 , 3 } } { R { n − 1 ,n } } { R 1 } { R 2 } { R n − 1 } { R n } . . . { L 1 } { L 2 } { L n − 1 } { L n } The graph ∆( PO n ) has 2 n maximal independent subsets. PO n has 2 n + 2 n − 2 maximal subsemigroups.

Order-preserving transformations � 2 n − 1 � |O n | = n { R { 1 , 2 } } { R { 2 , 3 } } { R { n − 1 ,n } } . . . { L 1 } { L 2 } { L n − 1 } { L n } The graph ∆( O n ) has A 2 n − 1 maximal independent subsets: A 1 = 1 , A 2 = A 3 = 2 , and A k = A k − 2 + A k − 3 for k > 3 . O n has A 2 n − 1 + 2 n − 4 maximal subsemigroups.

Order-preserving or -reversing partial transformations |POD n | = 2 |PO n | − n (2 n − 1) − 1 { R { 1 , 2 } , R { 6 , 7 } } { R { 2 , 3 } , R { 5 , 6 } } { R { 3 , 4 } , R { 4 , 5 } } { R 1 , R 7 } { R 2 , R 6 } { R 3 , R 5 } { R 4 } { L 1 , L 7 } { L 2 , L 6 } { L 3 , L 5 } { L 4 } The graph ∆( POD n ) has 2 ⌈ n/ 2 ⌉ maximal independent subsets. POD n has 2 ⌈ n/ 2 ⌉ + n − 1 maximal subsemigroups.

The Jones monoid 1 � 2 n � |J n | = C n = n + 1 n { R n } { R n − 1 } { R n − 2 } { R 2 } { R 1 } · · · { L n } { L n − 1 } { L n − 2 } { L 2 } { L 1 } Figure: The graph ∆( J n +1 ) . The graph ∆( J n +1 ) has 2 F n maximal independent subsets. J n +1 has 2 F n + 2 n − 1 maximal subsemigroups.

Summary: we’ve replicated previous results, and proved many new ones.