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

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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


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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

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Definitions

Alex McLeman Cayley Automaton Semigroups

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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

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Definitions cont.

Automata have outputs:

Alex McLeman Cayley Automaton Semigroups

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Definitions cont.

Automata have outputs:

  • q

x|y

  • r

Alex McLeman Cayley Automaton Semigroups

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Definitions cont.

Automata have outputs:

  • q

x|y

  • r

If we are in state q and read symbol x, we move to state r and

  • utput y. That is, δ(q, x) = (r, y).

Alex McLeman Cayley Automaton Semigroups

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Definitions cont.

Automata have outputs:

  • q

x|y

  • r

If we are in state q and read symbol x, we move to state r and

  • utput y. That is, δ(q, x) = (r, y).

If we’re in state q0 and read a sequence α1α2 . . . αn we output β1β2 . . . βn where δ(qi−1, αi) = (qi, βi).

Alex McLeman Cayley Automaton Semigroups

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Definitions cont.

Automata have outputs:

  • q

x|y

  • r

If we are in state q and read symbol x, we move to state r and

  • utput y. That is, δ(q, x) = (r, y).

If we’re in state q0 and read a sequence α1α2 . . . αn we output β1β2 . . . βn where δ(qi−1, αi) = (qi, βi). 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

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Definitions cont.

Automata have outputs:

  • q

x|y

  • r

If we are in state q and read symbol x, we move to state r and

  • utput y. That is, δ(q, x) = (r, y).

If we’re in state q0 and read a sequence α1α2 . . . αn we output β1β2 . . . βn where δ(qi−1, αi) = (qi, βi). 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

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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

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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

  • s

t|st

  • st

Alex McLeman Cayley Automaton Semigroups

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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

  • s

t|st

  • 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

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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

  • s

t|st

  • 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

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How does q act on S∗?

Alex McLeman Cayley Automaton Semigroups

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How does q act on S∗?

Let x ∈ S, α ∈ S∗, qi ∈ S. Then q · (xα) = (qx)(qx · α), (q1 · q2) · α = q1 · (q2 · α).

Alex McLeman Cayley Automaton Semigroups

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How does q act on S∗?

Let x ∈ S, α ∈ S∗, qi ∈ S. Then q · (xα) = (qx)(qx · α), (q1 · q2) · α = q1 · (q2 · α). For α = α1α2 . . . αn we have q · α = (qα1)(qα1 · α2 . . . αn) = (qα1)(qα1α2)(qα1α2 · α3 . . . α2) . . . = (qα1)(qα1α2) . . . (qα1 . . . αn)

Alex McLeman Cayley Automaton Semigroups

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How does q act on S∗?

Let x ∈ S, α ∈ S∗, qi ∈ S. Then q · (xα) = (qx)(qx · α), (q1 · q2) · α = q1 · (q2 · α). For α = α1α2 . . . αn we have q · α = (qα1)(qα1 · α2 . . . αn) = (qα1)(qα1α2)(qα1α2 · α3 . . . α2) . . . = (qα1)(qα1α2) . . . (qα1 . . . αn) So we can think of q as a function q : α1α2 . . . αn → (qα1)(qα1α2) . . . (qα1 . . . αn).

Alex McLeman Cayley Automaton Semigroups

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Some properties

Alex McLeman Cayley Automaton Semigroups

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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

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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

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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

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Zeros

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)).

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn.

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn. So z · a · α = z · β1β2 . . . βn = (z)n.

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn. So z · a · α = z · β1β2 . . . βn = (z)n. Consequently, Σ(C(Ln)) ∼ = Ln after noting y · α = (y)n = (z)n = z · α.

Alex McLeman Cayley Automaton Semigroups

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn. So z · a · α = z · β1β2 . . . βn = (z)n. Consequently, Σ(C(Ln)) ∼ = Ln 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

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn. So z · a · α = z · β1β2 . . . βn = (z)n. Consequently, Σ(C(Ln)) ∼ = Ln 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

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Zeros

Let z ∈ S be a left-zero. The z is a left-zero in Σ(C(S)). z · α = (zα1)(zα1α2) . . . (zα1 . . . αn) = (z)n.Let a ∈ S. Then a · α = β1β2 . . . βn. So z · a · α = z · β1β2 . . . βn = (z)n. Consequently, Σ(C(Ln)) ∼ = Ln 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)). Consider Rn. Then x · α = (xα1)(xα1α2) . . . (xα1 . . . αn) = α1α2 . . . αn and y · α = α1α2 . . . αn. So x = y but x = y.

Alex McLeman Cayley Automaton Semigroups

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When does x = y?

Alex McLeman Cayley Automaton Semigroups

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When does x = y?

Lemma Let x = y ∈ S. Then x = y ∈ Σ(C(S)) if and only if xa = ya for all a ∈ S.

Alex McLeman Cayley Automaton Semigroups

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When does x = y?

Lemma Let x = y ∈ S. Then x = y ∈ Σ(C(S)) if and only if xa = ya for all a ∈ S. Proof.

Alex McLeman Cayley Automaton Semigroups

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When does x = y?

Lemma Let x = y ∈ S. Then x = y ∈ Σ(C(S)) if and only if xa = ya for all a ∈ S. Proof. (⇒) Let aα ∈ S∗. Then x · aα = (xa)(xa · α) and y · aα = (ya)(ya · α). The first symbols of the outputs must be equal and so xa = ya for all a ∈ S.

Alex McLeman Cayley Automaton Semigroups

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When does x = y?

Lemma Let x = y ∈ S. Then x = y ∈ Σ(C(S)) if and only if xa = ya for all a ∈ S. Proof. (⇒) Let aα ∈ S∗. Then x · aα = (xa)(xa · α) and y · aα = (ya)(ya · α). The first symbols of the outputs must be equal and so xa = ya for all a ∈ S. (⇐) Let xa = ya. Then x · aα = (xa)(xa · α) = (ya)(ya · α) = y · aα and so x = y.

Alex McLeman Cayley Automaton Semigroups

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Nilpotent Semigroups

Alex McLeman Cayley Automaton Semigroups

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Nilpotent Semigroups

A semigroup S is nilpotent of class n if there exists n such that Sn = {0} and Sn−1 = {0}. Note that such a semigroup must necessarily contain a zero element. By definition a semigroup is nilpotent of class 1 if and only if it is trivial.

Alex McLeman Cayley Automaton Semigroups

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Nilpotent Semigroups

A semigroup S is nilpotent of class n if there exists n such that Sn = {0} and Sn−1 = {0}. Note that such a semigroup must necessarily contain a zero element. By definition a semigroup is nilpotent of class 1 if and only if it is trivial. Lemma (Cain 2009) Let S be finite and nilpotent of class n. Then Σ(C(S)) is finite and nilpotent of class n − 1.

Alex McLeman Cayley Automaton Semigroups

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Nilpotent Semigroups

A semigroup S is nilpotent of class n if there exists n such that Sn = {0} and Sn−1 = {0}. Note that such a semigroup must necessarily contain a zero element. By definition a semigroup is nilpotent of class 1 if and only if it is trivial. Lemma (Cain 2009) Let S be finite and nilpotent of class n. Then Σ(C(S)) is finite and nilpotent of class n − 1. Proof. We have w1 · w2 · . . . · wn−1 · α = (w1w2 . . . wn−1α1) . . . = 0ω since S is nilpotent of class n. Hence Σ(C(S)) is nilpotent of class at most n − 1.

Alex McLeman Cayley Automaton Semigroups

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Nilpotent Semigroups

A semigroup S is nilpotent of class n if there exists n such that Sn = {0} and Sn−1 = {0}. Note that such a semigroup must necessarily contain a zero element. By definition a semigroup is nilpotent of class 1 if and only if it is trivial. Lemma (Cain 2009) Let S be finite and nilpotent of class n. Then Σ(C(S)) is finite and nilpotent of class n − 1. Proof. We have w1 · w2 · . . . · wn−1 · α = (w1w2 . . . wn−1α1) . . . = 0ω since S is nilpotent of class n. Hence Σ(C(S)) is nilpotent of class at most n − 1. Now let w1, . . . , wn−1 be such that w1w2 . . . wn−1 = 0. Then w1 · . . . · wn−2 · wn−1 = (w1w2 . . . wn−2wn−1) = 0ω. Hence w1 · . . . · wn−2 = 0. So Σ(C(S)) is nilpotent of class n − 1.

Alex McLeman Cayley Automaton Semigroups

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Other known classes of Semigroups

Alex McLeman Cayley Automaton Semigroups

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Other known classes of Semigroups

Lemma (M 2012) Let S be cancellative (and not necessarily finite). Then Σ(C(S)) is free of rank equal to the order of S.

Alex McLeman Cayley Automaton Semigroups

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Other known classes of Semigroups

Lemma (M 2012) Let S be cancellative (and not necessarily finite). Then Σ(C(S)) is free of rank equal to the order of S. Lemma (M 2011) Let S be a finite monogenic semigroup with a non-trivial subgroup. Then Σ(C(S)) is a small extension of a free semigroup of rank equal to the order of the subgroup.

Alex McLeman Cayley Automaton Semigroups

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Other known classes of Semigroups

Lemma (M 2012) Let S be cancellative (and not necessarily finite). Then Σ(C(S)) is free of rank equal to the order of S. Lemma (M 2011) Let S be a finite monogenic semigroup with a non-trivial subgroup. Then Σ(C(S)) is a small extension of a free semigroup of rank equal to the order of the subgroup. Lemma (Maltcev 2008) Let S be finite. Then Σ(C(S)) is free if and only if the minimal ideal K of S consists of a single R-class in which every H-class is non-trivial and there exists k such that st = skt for all s, t ∈ S.

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples:

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band Left-zero semigroups

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band Left-zero semigroups Semilattices

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band Left-zero semigroups Semilattices Zero-unions of left-zero semigroups

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band Left-zero semigroups Semilattices Zero-unions of left-zero semigroups Ln ∪ B where Ln acts trivially on the band B

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups

S is self-automaton if S ∼ = Σ(C(S)). We are particularly interested in the map s → s. Known examples: A monoid is self automaton if and only if it is a band Left-zero semigroups Semilattices Zero-unions of left-zero semigroups Ln ∪ B where Ln acts trivially on the band B If S is regular and self-automaton then it is a band

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

Theorem Let B be a band. Then the map b → b is a homomorphism.

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

Theorem Let B be a band. Then the map b → b is a homomorphism. We can classify which bands are self-automaton.

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

Theorem Let B be a band. Then the map b → b is a homomorphism. We can classify which bands are self-automaton. Theorem (M 2012) Let B be a band. Then B ∼ = Σ(C(B)) under the map b → b if and

  • nly if the left-regular representation of B is faithful.

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

So are all self-automaton semigroups bands?

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

So are all self-automaton semigroups bands? NO!

Alex McLeman Cayley Automaton Semigroups

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Self-Automaton Semigroups cont

So are all self-automaton semigroups bands? NO! The semigroup defined by the following Cayley Table is not a band but is self-automaton: a b c d a b b b c b b b b b c c c c c d d d d d

Alex McLeman Cayley Automaton Semigroups

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An alternative construction

What if the states acted on the right of a sequence rather than the left? This is the approach taken by Cain.

Alex McLeman Cayley Automaton Semigroups

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An alternative construction

What if the states acted on the right of a sequence rather than the left? This is the approach taken by Cain. α · x = (xα1)(xα1α2)(xα1α2α3) . . . α · (x1 · x2) = (α · x1) · x2.

Alex McLeman Cayley Automaton Semigroups

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An alternative construction

What if the states acted on the right of a sequence rather than the left? This is the approach taken by Cain. α · x = (xα1)(xα1α2)(xα1α2α3) . . . α · (x1 · x2) = (α · x1) · x2. Denote the semigroup generated by the states with this right action by Π(C(S)).

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

Cain conjectures the following: Conjecture S ∼ = Π(C(S)) if and only if S is a band in which every D-class is square and every maximal D-class is a singleton.

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

Cain conjectures the following: Conjecture S ∼ = Π(C(S)) if and only if S is a band in which every D-class is square and every maximal D-class is a singleton. How does this right action construction relate to the previously defined left actions?

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

Cain conjectures the following: Conjecture S ∼ = Π(C(S)) if and only if S is a band in which every D-class is square and every maximal D-class is a singleton. How does this right action construction relate to the previously defined left actions? Theorem S ∼ = Π(C(S)) if and only if S is self-dual and S ∼ = Σ(C(S)).

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

To tackle Cain’s conjecture we should look at self-dual self-automaton semigroups.

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

To tackle Cain’s conjecture we should look at self-dual self-automaton semigroups. Theorem (M 2013) Let S be self-dual and self-automaton. If S2 = S then S is a band.

Alex McLeman Cayley Automaton Semigroups

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An alternative construction cont

To tackle Cain’s conjecture we should look at self-dual self-automaton semigroups. Theorem (M 2013) Let S be self-dual and self-automaton. If S2 = S then S is a band. A complete classification of self-automaton semigroups (both self-dual and otherwise) remains an open question.

Alex McLeman Cayley Automaton Semigroups

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Thanks for listening!

Alex McLeman Cayley Automaton Semigroups