Universal sequences for groups and semigroups

Transcription

1 Universal sequences for groups and semigroups J. D. Mitchell (joint work with J. Hyde, J. Jonušas, and Y. Péresse) School of Mathematics and Statistics, University of St Andrews 12th September 2017 J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

2 Universal words Let F (A) be a free group over some alphabet A and let G be a group. A word w F (A) is universal for G if for every g G there exists a homomorphism φ : F (A) G such that (w)φ = g. Theorem (Oré 51) Every element of the symmetric group S X on an infinite set X is a commutator, i.e. a 1 b 1 ab = [a, b] is universal for S X. Example The word a G +1 is universal for any finite group G. The word a 2 is not universal for S X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

3 More groups... Theorem (Truss 0X) Let G be any Polish group with a comeagre conjugacy class. Then every element of G is a commutator. Proof. let C = C 1 be a comeagre conjugacy class of G Cg is comeagre for all g G Cg C (Baire s Category Theorem) if a Cg C, then g = ba for some b C a 1 and b are conjugate and so there exists c G such that c 1 a 1 c = b and so g = c 1 a 1 ca = [c, a]. For example, G might be Aut(R), Aut(Q, ), F (A) when A is countable, the isometries of the Urysohn space, homeomorphisms of the Cantor space,... J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

4 More words... Theorem (Lyndon 90, Dougherty and Mycielski 99) Let F (A) be any free group, let w F (A), and let X be any infinite set. Then w is universal for every element of S X if and only if w is not a proper power of any other word in F (A). We denote by R the countable random graph. Theorem (Droste-Truss 06) Let F (A) be any free group and let w F (A). Then w is universal for every special element of Aut(R) if and only if w is not a proper power of any other word in F (A). J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

5 From groups to semigroups If A is a set, then A + denotes the free semigroup consisting of all words over A. Let S be a semigroup and let A be any alphabet. Then a word w A + is universal for S if for any s S there exists a homomorphism φ : A + S such that (w)φ = s. If X is any set, then the set X X of functions from X to X is a semigroup under the usual composition of functions. Theorem (Cayley s theorem) Every semigroup is isomorphic to a subsemigroup of some X X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

6 Groups versus semigroups Words over the free group can contain inverses, but words over the free semigroup cannot. The free semigroup A + is a subsemigroup of the free group F (A) on A, hence any universal word in A + for a group is also universal in F (A). On the other hand, every universal word in F (A) for a group G is universal for G in (A A 1 ) +. So, these notions are not really distinct. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

7 Universal words for X X Hyde and Jonušas Example The word abc is universal for X X. The word a n, n N, is universal for X X if and only if n = 1. Theorem (Isbell 66, Hyde and Jonušas 14) Let A be any alphabet, and let w A + be such that every proper prefix of w is not a proper suffix of w. Then w is universal for X X. Example The words b 3 ab 2 a 6 and b 3 ab 2 a 6 d 3 c are universal for X X. The word aba 2 b 2 ab is also universal for X X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

8 From one word to many... Let S be a semigroup and let A be any alphabet. Then a sequence of words w 0, w 1,... A + is universal for S if for any sequence s 0, s 1,... S there exists a homomorphism φ : A + S such that (w n )φ = s n for all n N. Theorem (Schreier-Ulam 34, Sierpiński 34) The sequences (ab n 1 cd n 1 e) n N and (ab n 1 cd n 1 ) n N are universal for the semigroup of continuous functions on [0, 1] R. Theorem (Sierpiński 35) The sequence (a 2 b 3 (abab 3 ) n+1 ab 2 ab 3 ) n N is universal for X X for any infinite set X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

9 The Hyde-Star-Trek-Analogy proof... Theorem (Sierpiński 35) The sequence (a(ab) n b) n N is universal for X X for any infinite set X. Proof. Suppose X is countable and identify X with the eventually constant sequences over N. let f 1, f 2,... X X define: a : (x 0, x 1,...) (0, x 0, x 1,...) b : (x 0, x 1,...) (x 0 + 1, x 1,...) c : (x 0, x 1,...) (x 1, x 2,...)f x0 (x 0, x 1,...) a (0, x 0, x 1,...) bn (n, x 0, x 1,...) c (x 0, x 1,...)f n the previous line implies that: ab n c = f n for all n 1 if we set f 0 = b, then b = f 0 = ac and so f n = a(ac) n c. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

10 Universal sequences for X X Some sequences of universal words for X X : (a 2 b 3 (abab 3 ) n+1 ab 2 ab 3 ) n N (Sierpiński) (aba n+1 b 2 ) n N (Banach) (abab n+3 ab 2 ) n N (Hall) (aba n+2 b n+2 ) n N (Mal cev) (a 2 b n+2 ab) n N (McNulty) The monoid Surj(X) of surjective functions on X where X = ℵ 5 has the property that every countable subset is contained in a 42-generated subsemigroup but Surj(X) has no universal sequences. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

11 Universal sequences for groups Galvin showed that (a 1 (a n ba n )b 1 (a n b 1 a n )ba) n N is universal for S X. If G is the group of homeomorphisms of the Cantor space, Q, or R \ Q, then (a 1 (a n ba n )b 1 (a n b 1 a n )ba) n N is universal for G. Hyde-Jonušas-M-Péresse showed that 3n(n+1) 1 i=3n(n 1) is universal for Aut(Q, ). [a b 8i 1, a b8i+2c ] d2 [a b 8i 3, a b8i+4c ] d4 [a b 8i 5, a b8i+6c ] d3 [a b 8i 7, a b8i+8c ] d5 Hyde-Jonušas showed that every finite sequence (a bn 1 c dn 1 ) 1 n N, N N is universal for the automorphism group of the random graph. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23 n N

12 Universal sequences for inverse semigroups A semigroup S is inverse if for all x S there exists a unique y S such that xyx = x and yxy = y. If X is any set, then the set I X of bijections between subsets of X is an inverse semigroup under the usual composition of binary relations; called the symmetric inverse monoid. Theorem (Wagner-Preston representation heorem) Every inverse semigroup is isomorphic to an inverse subsemigroup of some I X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

13 Universal sequences for the symmetric inverse monoid Theorem (Hyde-Jonušas-M-Péresse) Let S be a semigroup with identity 1 S whose group of units contains a subgroup G such that G D embeds in G. Suppose that some other conditions hold. Then a 3 (ab) n ba(ab) n (bab) 3 is universal for S. Corollary Let X be an infinite set. Then the sequence a 3 (ab) n ba(ab) n (bab) 3 is universal for the: (i) symmetric inverse monoid I X ; (ii) the partition monoid P X ; and (iii) dual symmetric inverse monoid I X when X 2ℵ 0. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

14 Clones and terms Let X be an infinite set. Then X X X is closed under superposition: if f, g X X X, then f(g(x, y), y), f(x, g(x, y)) X X X. Instead of words, we now have terms such as g(g(g(x, g(x, y)), g(x, g(x, y))), g(x, g(x, y))). Theorem ( Los 1950) Let f 0, f 1,... X X X. Then there exists g X X X such that f n is a finite superposition of g for all n N. Corollary There is an infinite universal set of terms for X X X. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

15 The first term in the sequence f 0 (x, y) = g(g(g(g(g(g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))))), g(g(g(g(x, g(x, x )), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g (x, g(x, x)), g(x, g(x, x))), g(x, g(x, x)))))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x)))))), g(g(g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))))), g(g (g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g( x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x)))))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))))))), g(g(g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g (x, x))))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g( x, x))), g(x, g(x, x))), g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x)))))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g(g(g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))), g(g (g(x, g(x, x)), g(x, g(x, x))), g(x, g(x, x))))))), g(g(g(g(g(g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g (y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g(g(g(g(g(y, g (y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g (g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y ))), g(y, g(y, y))))))), g(g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g (y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g (g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g( y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g(g(g(g(y, g(y, y)), g(y, g (y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y )), g(y, g(y, y))), g(y, g(y, y)))))))), g(g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y)) ), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g(g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g( y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y)))))))), g(g(g(g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(g(y, g(y, y)), g(y, g(y, y))), g(y, g(y, y))), g(g(g(y, g(y, y)) J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

16 Cardinalities Proposition If S is a non-trivial semigroup and S has a universal sequence over finite alphabet, then S 2 ℵ 0. So, for example, no non-trivial finitely generated group has a universal sequence. Corollary Groups with a universal sequence over a 1-letter alphabet do not exist. Proof. Suppose G is a group with a 1-letter universal sequence. Then: every countable subset of G is contained in a cyclic subgroup G = Z and G < 2 ℵ 0, a contradiction. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

17 Bergman s Property A semigroup S has Bergman s property if given any generating set U for S there exists n N such that S = U U 2 U n. Theorem (Bergman 05) The symmetric group on any infinite set has my property! Many groups have Bergman s property: (Shelah 76) there exists a group G with U 240 = G for all generating sets U (Droste & Göbel 05) the order-automorphisms of the rationals, the homeomorphisms of the irrationals,... Some do not: free groups and finitely generated infinite groups (Droste & Göbel 05) {f S Q : k, x Q, x (x)f k} J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

18 Universal sequence implies Bergman s property Proposition (Cornulier 08, Khelif 06) If S is any semigroup with a universal sequence over a finite alphabet, then S has Bergman s property. So, Bergman s theorem is a consequence of Galvin s proof that (a 1 (a n ba n )b 1 (a n b 1 a n )ba) n N is universal for S X for any infinite set X. What about the converse? Does every semigroup/group with Bergman s property have a universal sequence over a finite alphabet? If S is any countable set with multiplication xy = x for all x, y S, then S is a semigroup with the unique generating set S, and hence has BP, but it has not universal sequences. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

19 Language theoretic properties Proposition Let G be a group and let (w n ) n N be a universal sequence for G over some alphabet. Then {w n : n N} A is not a context-free language. The sequences (a 3 (ab) n ba(ab) n (bab) 3 ) n N is universal for the symmetric inverse monoid, and (a 3 (ab) n ba(ab) n (bab) 3 ) n N is context-free. Proposition Let S be a non-trivial inverse semigroup and let (w n ) n N be a universal sequence for S over some alphabet. Then {w n : n N} A is not a regular language. The sequence (a(ab) n b) n N is universal for X X and (a(ab) n b) n N is regular. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

20 Change of cardinality Is it possible that for some X the sequence (w n ) n N is universal for S X but for some Y with Ω Σ, (w n ) n N is not universal for S Y? Theorem (Hyde-Jonušas-M-Péresse) Let X and Y be infinite sets such that X < Y. Then every sequence that is universal for S X is universal for S Y. If X is a regular cardinal and 2 ℵ 0 < X < Y, then every sequence that is universal for S Y is universal for S X. Proposition Let S be a semigroup and let α be some cardinal number. Then a sequence is universal for S if and only if it is universal for S α. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

21 Theorem Let X and Y be infinite sets such that X < Y. Then every sequence that is universal for S X is universal for S Y. Proof. Suppose (w n ) n N is universal for S X and (g n ) n N is a sequence in S Y Partition the orbits of G := g n : n N on Y into sets X y such that there exists a bijection µ y : X y X for all y Y f : G SX Y g (µ 1 y gµ y ) y Y f : SX Y S Y (h y ) y Y y Y µ yh y µ 1 y are monomorphisms and (g)ff = g for all g G (w n ) n N is universal for SX Y and so there is a homomorphism φ : A + SX Y such that (w n)φ = (g n )f for all n N φ f : A + S Y is a homomorphism such that (w n )φ = g n for all n N. J. D. Mitchell (St Andrews) Universal sequences 12th September / 23

22 Problem Does there exist a universal sequence for Aut(R)? If S is a semigroup with Bergman s property, and T is a subsemigroup of S with S \ T <, then does T have BP also? If X and Y are infinite sets, then is every universal sequence for S X also universal for S Y? Thanks! J. D. Mitchell (St Andrews) Universal sequences 12th September / 23