Cayley Automaton Semigroups Alex McLeman alexm@mcs.st-andrews.ac.uk - PowerPoint PPT Presentation

Cayley Automaton Semigroups Alex McLeman alexm@mcs.st-andrews.ac.uk The 4th Novi Sad Algebraic Conference - Semigroups and Applications - 7th June 2013 Alex McLeman Cayley Automaton Semigroups

Definitions Alex McLeman Cayley Automaton Semigroups

Definitions Definition An automaton is a triple A = ( Q , B , δ ) where: Q is a finite set of states B is a finite alphabet δ : Q × B → Q × B is the transition function. Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: ���� ���� x | y � ���� ���� q r Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: ���� ���� x | y � ���� ���� q r If we are in state q and read symbol x , we move to state r and output y . That is, δ ( q , x ) = ( r , y ) . Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: ���� ���� x | y � ���� ���� q r If we are in state q and read symbol x , we move to state r and output y . That is, δ ( q , x ) = ( r , y ) . If we’re in state q 0 and read a sequence α 1 α 2 . . . α n we output β 1 β 2 . . . β n where δ ( q i − 1 , α i ) = ( q i , β i ) . Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: ���� ���� x | y � ���� ���� q r If we are in state q and read symbol x , we move to state r and output y . That is, δ ( q , x ) = ( r , y ) . If we’re in state q 0 and read a sequence α 1 α 2 . . . α n we output β 1 β 2 . . . β n where δ ( q i − 1 , α i ) = ( q i , β i ) . Starting in state q and reading α gives an endomorphism of the | B | -ary rooted tree. Extending this to several states gives a homomorphism φ : Q + → End ( B ∗ ) . Alex McLeman Cayley Automaton Semigroups

Definitions cont. Automata have outputs: ���� ���� x | y � ���� ���� q r If we are in state q and read symbol x , we move to state r and output y . That is, δ ( q , x ) = ( r , y ) . If we’re in state q 0 and read a sequence α 1 α 2 . . . α n we output β 1 β 2 . . . β n where δ ( q i − 1 , α i ) = ( q i , β i ) . Starting in state q and reading α gives an endomorphism of the | B | -ary rooted tree. Extending this to several states gives a homomorphism φ : Q + → End ( B ∗ ) . We say that Σ( A ) ∼ = im ( φ ) is the automaton semigroup. Alex McLeman Cayley Automaton Semigroups

Cayley Automaton Semigroups C ( S ) is the automaton arising from the Cayley Table of S . Each element s ∈ S gives a state s . Transitions are defined by right-multiplication in S : reading symbol t in state s moves us to state st and outputs symbol st . Alex McLeman Cayley Automaton Semigroups

Cayley Automaton Semigroups C ( S ) is the automaton arising from the Cayley Table of S . Each element s ∈ S gives a state s . Transitions are defined by right-multiplication in S : reading symbol t in state s moves us to state st and outputs symbol st . A typical edge looks like ���� ���� t | st � ���� ���� s st Alex McLeman Cayley Automaton Semigroups

Cayley Automaton Semigroups C ( S ) is the automaton arising from the Cayley Table of S . Each element s ∈ S gives a state s . Transitions are defined by right-multiplication in S : reading symbol t in state s moves us to state st and outputs symbol st . A typical edge looks like ���� ���� t | st � ���� ���� s st More formally: C ( S ) = ( S , S , δ ) , δ ( s , t ) = ( st , st ) where we denote states by s to avoid confusion. Alex McLeman Cayley Automaton Semigroups

Cayley Automaton Semigroups C ( S ) is the automaton arising from the Cayley Table of S . Each element s ∈ S gives a state s . Transitions are defined by right-multiplication in S : reading symbol t in state s moves us to state st and outputs symbol st . A typical edge looks like ���� ���� t | st � ���� ���� s st More formally: C ( S ) = ( S , S , δ ) , δ ( s , t ) = ( st , st ) where we denote states by s to avoid confusion. Σ( C ( S )) is the Cayley Automaton Semigroup. Alex McLeman Cayley Automaton Semigroups

How does q act on S ∗ ? Alex McLeman Cayley Automaton Semigroups

How does q act on S ∗ ? Let x ∈ S , α ∈ S ∗ , q i ∈ S . Then q · ( x α ) = ( qx )( qx · α ) , ( q 1 · q 2 ) · α = q 1 · ( q 2 · α ) . Alex McLeman Cayley Automaton Semigroups

How does q act on S ∗ ? Let x ∈ S , α ∈ S ∗ , q i ∈ S . Then q · ( x α ) = ( qx )( qx · α ) , ( q 1 · q 2 ) · α = q 1 · ( q 2 · α ) . For α = α 1 α 2 . . . α n we have q · α = ( q α 1 )( q α 1 · α 2 . . . α n ) = ( q α 1 )( q α 1 α 2 )( q α 1 α 2 · α 3 . . . α 2 ) . . . = ( q α 1 )( q α 1 α 2 ) . . . ( q α 1 . . . α n ) Alex McLeman Cayley Automaton Semigroups

How does q act on S ∗ ? Let x ∈ S , α ∈ S ∗ , q i ∈ S . Then q · ( x α ) = ( qx )( qx · α ) , ( q 1 · q 2 ) · α = q 1 · ( q 2 · α ) . For α = α 1 α 2 . . . α n we have q · α = ( q α 1 )( q α 1 · α 2 . . . α n ) = ( q α 1 )( q α 1 α 2 )( q α 1 α 2 · α 3 . . . α 2 ) . . . = ( q α 1 )( q α 1 α 2 ) . . . ( q α 1 . . . α n ) So we can think of q as a function q : α 1 α 2 . . . α n �→ ( q α 1 )( q α 1 α 2 ) . . . ( q α 1 . . . α n ) . Alex McLeman Cayley Automaton Semigroups

Some properties Alex McLeman Cayley Automaton Semigroups

Some properties (Mintz 2009) Let S be finite. The following are equivalent: S is aperiodic Σ( C ( S )) is finite Σ( C ( S )) is aperiodic Alex McLeman Cayley Automaton Semigroups

Some properties (Mintz 2009) Let S be finite. The following are equivalent: S is aperiodic Σ( C ( S )) is finite Σ( C ( S )) is aperiodic (Silva and Steinberg 2005) Let G be a non-trivial finite group. Then Σ( C ( G )) ∼ = F | G | Alex McLeman Cayley Automaton Semigroups

Some properties (Mintz 2009) Let S be finite. The following are equivalent: S is aperiodic Σ( C ( S )) is finite Σ( C ( S )) is aperiodic (Silva and Steinberg 2005) Let G be a non-trivial finite group. Then Σ( C ( G )) ∼ = F | G | (Mintz 2009) Let T ≤ S . The Σ( C ( T )) divides Σ( C ( S )) . If T is a non-trivial group then Σ( C ( T )) ≤ Σ( C ( S )) . Alex McLeman Cayley Automaton Semigroups

Zeros Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Let a ∈ S . Then a · α = β 1 β 2 . . . β n . Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Let a ∈ S . Then a · α = β 1 β 2 . . . β n . So z · a · α = z · β 1 β 2 . . . β n = ( z ) n . Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Let a ∈ S . Then a · α = β 1 β 2 . . . β n . So z · a · α = z · β 1 β 2 . . . β n = ( z ) n . Consequently, Σ( C ( L n )) ∼ = L n after noting y · α = ( y ) n � = ( z ) n = z · α. Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Let a ∈ S . Then a · α = β 1 β 2 . . . β n . So z · a · α = z · β 1 β 2 . . . β n = ( z ) n . Consequently, Σ( C ( L n )) ∼ = L n after noting y · α = ( y ) n � = ( z ) n = z · α. Let 0 ∈ S be the zero element. Then 0 is the zero element in Σ( C ( S )) . Alex McLeman Cayley Automaton Semigroups

Zeros Let z ∈ S be a left-zero. The z is a left-zero in Σ( C ( S )) . z · α = ( z α 1 )( z α 1 α 2 ) . . . ( z α 1 . . . α n ) = ( z ) n . Let a ∈ S . Then a · α = β 1 β 2 . . . β n . So z · a · α = z · β 1 β 2 . . . β n = ( z ) n . Consequently, Σ( C ( L n )) ∼ = L n after noting y · α = ( y ) n � = ( z ) n = z · α. Let 0 ∈ S be the zero element. Then 0 is the zero element in Σ( C ( S )) . Let z ∈ S be a right zero. Then z is a right-zero in Σ( C ( S )) . Alex McLeman Cayley Automaton Semigroups