Rewriting Generators for (Sub)Semigroups Nik Ruskuc [email protected] School of Mathematics and Statistics, University of St Andrews Porto, 6 July 2009 Setting the Scene S { a semigroup/monoid S = hAi T ≤ S Find a generating set for T . University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups People I Robert Gray, NR, Green index and finiteness conditions for semigroups, J. Algebra 320 (2008), 3145{3164. I Alan J. Cain, Robert Gray, NR, Green index in semigroups: generators and related finiteness conditions, about to be submitted. I Also: C.M. Campbell, M. Hoffmann, E.F. Robertson, R.M Thomas. I Related: R. Gray, V. Maltcev, J.D. Mitchell, NR, Ideals, finiteness conditions and Green index for subsemigroups, about to be submitted. A first naive approach Given a `long' word from A∗ representing an element of T , `decompose' it into `shorter' members of T . a1a2a3ja4a5ja6ja7a8a9j ::: 62 T 2 T Example S = fa; bg∗ T = fwords of even lengthg abjaajbajbb Hence T = ha2; ab; ba; b2i. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Variation: finite complement (Rees index) Suppose jS n T j = r < 1 Consider a product a1a2 ::: ar+1. Then: (i) 9k : a1 ::: ak 2 T , or (ii) 9k; l : a1 ::: ak = a1 ::: al . Theorem (Jura 1978) If S = hAi then T = hfuav : u; v 2 (S n T )1; a 2 A; ua; uav 2 T gi: Corollary S is finitely generated if and only if T is finitely generated. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Rewriting mapping Suppose S = hAi, T ≤ S, T = hBi. L(A; T ) = fwords from A∗ representing elements of T g Then there exists a mapping φ : L(A; T ) ! B∗ such that w 2 L(A; T ) and φ(w) 2 B∗ represent the same element of T . From the previous slides: For w 2 L(A; T ) let w = uav, where ua is the shortest prefix in L(A; T ). Then (ua) · φ(v) if v 2 L(A; T ) φ(w) = w otherwise: University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Another idea: `Coset representatives' S S S S h1 h1 h1 u u =t h 2 h2 h2 h2 t fhi : i 2hI3g h3 h3 hi s = τ(i; s)hλ(i;s) τ(i; s) 2 T ; λ(i; s) 2 I h4 h4 h4 h5 h5 h5 h6 h6 h6 University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups hi s = τ(i; s)hλ(i;s) 1 a1a2a3 ::: ak = τ(i1; a1)hλ(i1;a1)a2a3 ::: ak = ::: = τ(i1; a1)τ(i2; a2) : : : τ(ik ; ak )hik+1 `Problem': hik+1 need not belong to T . University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Groups: Schreier Theorem G = hAi { a group H ≤ G fhi : i 2 I g { left coset representatives. Theorem (Schreier 1927) H = hfτ(i; a): i 2 I ; a 2 Agi. Corollary G is finitely generated if and only if H is finitely generated. Rewriting mapping: φ(i; aw) = τ(i; a)φ(λ(i; a); w) φ(w) = φ(1; w): University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups H-classes Exactly the same works in monoids for: I maximal subgroups (group H-classes); I arbitrary subgroups; I Sch¨utzenberger groups. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Relative Green's relations (Wallace 1963) S T u LT v , Su = Sv u RT v , uS = vS u HT v , u LT v & u RT v (T ≤ S) Definition T The Green index [S : T ]G of T is S is the number of H -classes in S n T . University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Green, Rees and group indices Proposition (i) jS n T j < 1 ) [S : T ]G < 1. (ii)[ G : H] < 1 , [G : H]G < 1. Proof (ii) LH -classes = right cosets RH -classes = left cosets HH -classes = intersections University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Green rewriting hi s = τ(i; s)hλ(i;s) T T fhi : i 2 I g { representatives of L -classes, or H -classes. hi1 a1a2a3 ::: ak = τ(i1; a1)hi2 a2a3 ::: ak = ::: = τ(i1; a1)τ(i2; a2) : : : τ(ik ; ak )hik+1 h s 2 H if h s 2 S n T λ(i; s): i λ(i;s) i 1 otherwise University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Example Set: S = Z2 × Z × Z2 [ f0g (a; b + e; f ) if c = d = 0 Multiplication: (a; b; c)(d; e; f ) = 0 otherwise T = f(a; b; c): a ≥ cg [ f0g S n T = f(0; x; 1) : x 2 Zg (0; y − x; 0)(0; x; 1) = (0; y; 1) Hence S n T is an LT -class. T is not finitely generated, as all (1; x; 1) are indecomposable. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Green rerewriting hi s = τ(i; s)hλ(i;s) a1a2a3 ::: ak = τ(i1; a1)τ(i2; a2) : : : τ(ik ; ak )hik+1 shi = hρ(s;i)σ(s; i) τ(i1; a1)τ(i2; a2) : : : τ(ik ; ak )hik+1 = τ(i1; a1) : : : τ(ik−1; ak−1) hρ(τ(ik ;ak );jk ) σ(τ(ik ; ak ); jk ) = ::: = hj0 σ(τ(i1; a1); j1) : : : σ(τ(ik−1; ak−1); jk−1)σ(τ(ik ; ak ); jk ) Lemma If a1 ::: ak 2 T then hj0 = 1. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Generation theorems for Green index Theorem T If S = hAi and fhi : i 2 I g are representatives of H -classes in S n T then T = hfσ(τ(i; a); j): a 2 A; i; j 2 I gi: Corollary Suppose [S : T ]G < 1. Then S is finitely generated if and only if T is finitely generated. Rewriting mapping: φ(a1a2 ::: ak ) = bi1;a1;j1 bi2;a2;j2 ::: bik ;ak ;jk + recursive formulas for il , jl . University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Do we need HT -classes? Or is sufficient to require: S 0 0 h 8s 2 S : 9i 2 I : 9t; t 2 T : s = thi = hi t ? 1 u =t h 2 h2 t h3 h4 h5 h6 Do we need HT -classes? S 1 Example M M { a f.g. monoid M J { a non f.g. ideal M = fm : m 2 Mg S = M[· M mn = mn = mn _ 1 S = hM [ f1M gi M _ T = M [ J M T not f.g. But: m = 1M m = m1M What are rewriting mappings for? Typically, they are involved in statements and proofs regarding properties and conditions related to generators. For example. Theorem Suppose S is defined by a presentation hA j Ri, and that T = hBi ≤ S. Furthermore, suppose φ : L(A; T ) ! B∗ is a rewriting mapping. Then the presentation hB j φ(β) = b; φ(w1w2) = φ(w1)φ(w2); φ(w3uw4) = φ(w3vw4) (b 2 B; w1; w2; w3uw4 2 L(A; T ); (u; v) 2 R)i defines T . University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Reidemeister{Schreier type theorems Theorem Suppose [G : H] < 1. Then G is finitely presented if and only if H is finitely presented. Proof Recall: φ(i; aw) = τ(i; a)φ(λ(i; a); w), φ(w) = φ(1; w). φ(w1w2) = φ(w1)φ(w2) φ(w3uw4) = φ(w3)φ(λ(1; w3); u)φ(λ(1; w3u); w4) Theorem Suppose jS n T j < 1. Then S is finitely presented if and only if T is finitely presented. Proof Similar but much harder. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Reidemeister{Schreier{Green? And if it [the solution] was perhaps less sound than I had thought in the first flush of discovery, its inelegance never diminished. And it was above all inelegant in this, to my mind. (S. Beckett, Molloy) Problem If S is finitely presented and [S : T ]G < 1 is T finitely presented? What is at issue: The way φ is defined (re-re-writing). Theorem Suppose [S : T ]G < 1. If T and all the relative Sch¨utzenberger groups are finitely presented then S is finitely presented. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups But: Reidemeister{Schreier{Malcev{Green Definition Suppose S is embeddable into a group. A presentation hA j Ri is a Malcev presentation for S if S =∼ A∗/ρ where ρ is the smallest congruence on A∗ such that A∗/ρ is embeddable into a group. Theorem Suppose [S : T ]G < 1. Then S has a finite Malcev presentation if and only if T has a finite Malcev presentation. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Automatic structures and rewriting Obervation All the rewriting mappings we have encountered so far can be realised by transducers. Theorem If [S : T ]G < 1 and S is automatic then T is automatic. Remark The converse holds for groups and finite Rees index, but not in general. University of St Andrews Nik Ruskuc: Rewriting for (sub)semigroups Syntactic congruences ρ { an equivalence on a semigroup S Σ(ρ) { the largest congruence on S contained in ρ Σ(T ) (for T ≤ S) = Σ((T × T ) [ ((S n T ) × (S n T ))) Example If G is a group and H ≤ G then Σ(H) is the congruence determined by the kernel of H. Example If S is a semigroup and J is an ideal, then Σ(J) is the Rees congruence (J × J) [ ∆S .