Automatic Presentations and Classes of Semigroups Graham Oliver - - PDF document

Automatic Presentations and Classes of Semigroups Graham Oliver University of Leicester

Joint work with Prof. Rick Thomas

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Automatic Presentations

• Finite presentations of infinite structures
• Generic approach to deciding FO theory
• Restriction of recursive structures
• Inspired by theory of automatic groups

and automatic semigroups

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Automatic Presentations

x = x1x2 . . . xr y = y1y2 . . . ys |x| ≤ |y| Convolution of x and y, conv(x, y), is:

• x1

y1 x2 y2

• . . .
• xr

yr

• yr+1
• . . .
• ys
• An Automatic Presentation for

a structure (S, R1, . . . , Rn) consists of:

• Regular language L ⊆ Σ∗
• Surjective map: θ : L → S
• L= = {conv(x1, x2) : θ(x1) = θ(x2)}

is regular

• LRi = {conv(x1, . . . , xk) : Ri(θ(x1), . . . , θ(xk)}

is regular

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Automatic Presentations

For a structure S with an automatic presenta- tion:

• The FO-theory of S is decidable
• If T is FO-interpretable in S then T has an

automatic presentation

• S is FO-interpretable in (N, +, |2)
• S is FO-interpretable in ({0, 1}⋆, , P0, P1, el)

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Semigroups

(S, ◦), where ◦ is an associative binary function.

• 1. Examples:

(a) Σ = {a, b}, (Σ⋆◦) where e.g. aba ◦ bbb = ababbb (b) (N, ◦), (Z, ◦), (Q, ◦), (R, ◦), (C, ◦) where ◦ ∈ {+, ×} (c) (Partial) Automorphisms

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Groups

• 1 ◦ s = s ◦ 1 = s
• ∃s−1, s ◦ s−1 = 1 = s−1 ◦ s

A group is virtually abelian if it contains an abelian subgroup of finite index. Theorem (G.O, R.Thomas STACS05): A f.g. group G has an automatic pre- sentation if and only if G is virtually abelian Corollary: The class of f.g. groups with automatic presentations is properly contained in the class of automatic groups.

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Commutative Semigroups

• x ◦ y = y ◦ x

Theorem (Taitslin): All f.g. commutative semigroups are FO-interpretable in (N, +). Corollary: All f.g. commutative semigroups have automatic presentations. Note: There exists a f.g. commutative semigroup that is not automatic. (Hoffmann, Thomas)

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Cancellative Semigroups

• a ◦ x = a ◦ y ⇒ x = y
• x ◦ b = y ◦ b ⇒ x = y

Proposition (G.O, R.Thomas): If a f.g. semigroup S has an automatic presentation, then S has polynomial growth Let GS = {s−1 ◦ t : s, t ∈ S}. If GS is a group, it is called the group of (left) quotients of S. Proposition (Grigorchuk): If a f.g. cancellative semigroup S has polynomial growth, then S has a group of (left) quotients GS

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Cancellative Semigroups

S - semigroup, GS - group of (left) quotients Proposition: GS is FO-interpretable in S FO-interpretation: θ(x, y) :=∀z, z = z f : S2 →G f(s, t) =s−1 ◦ t θ=(x1, y1; x2, y2) := ∃p, q(x1 ◦ p = y1 ◦ q ∧ x2 ◦ p = y2 ◦ q) θ◦(x1, y1; x2, y2; x3, y3) := ∃p, q(x3 = p ◦ x1 ∧ y3 = q ◦ y2 ∧ q ◦ x2 = p ◦ y1)

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Cancellative Semigroups

Theorem: Let S be a f.g. cancellative semigroup with an automatic presentation; then, S embeds in a virtually abelian group. Proof: The group of (left) quotients of S, GS, has an automatic presentation; so, GS is virtually abelian. Conjecture: A f.g. cancellative semigroup S has an automatic presentation if and only if S embeds in a virtually abelian group.

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Conclusion

Theorem: A f.g. group G has an automatic pre- sentation if and only if G is virtually abelian Theorem: All f.g. commutative semigroups have automatic presentations. Theorem: Let S be a f.g. cancellative semigroup with an automatic presentation; then, S embeds in a virtually abelian group.

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