Approaches to tiling semigroups

Transcription

1 Nick Gilbert Heriot-Watt University, Edinburgh (joint work with Erzsi Dombi)

2 Periodic tilings

3 Non-periodic tilings Ulrich Kortenkamp: Paving the Alexanderplatz

4 Tilings of R n A tile in R n is a connected bounded subset t that is the closure of its interior, with a colour assigned from a set Σ. A tiling T of R n is a union of tiles meeting only at their boundaries. We always assume there are only finitely many different types of tiles: type equivalence up to Σ-preserving translation in R n. One tile type and one colour give a vanilla tiling of R n. A marked pattern in T is a finite connected collection of tiles in T, with two chosen distinguished tiles, the in-tile and the out tile (possibly equal) up to equivalence under translation.

5 History Johannes Kellendonk (1997): intro tiling semigroups Kellendonk & Mark Lawson (2000): general theory Yongwen Zhu (2002): algebraic properties Lawson (2004): specific theory for one-dimensional case Alan Forrest, John Hunton & Kellendonk ( ): cohomological invariants Don McAlister & Filipa Soares ( ) Erzsi Dombi & NDG ( )

6 The tiling semigroup tiling T has tiling semigroup S(T ), with 0 S(T ), non-zero elements are marked patterns, multiply P, Q by matching out-tile of P with in-tile of Q: if resultant overlap matches, and then if P Q is a pattern, PQ = P Q marked at the in-tile of P and the out-tile of Q. Non-matching = PQ = 0. Three possible reasons for PQ = 0: in- and out- tiles don t match, rest of overlap doesn t match, P Q isn t a pattern in T

7 Multiplication in S(T )

8 Basic properties of tiling semigroups idempotent in-tile = out-tile tiling semigroups are inverse: get P 1 by switching the inand out- tiles tiling semigroups are combinatorial, completely semisimple, E unitary inverse semigroups. natural partial order is reverse inclusion of marked patterns: P Q P Q as marked patterns so big patterns are below small ones in the natural partial order.

9 One dimensional tilings Fix a finite alphabet A = {a 1,..., a n } (n 2 most of the time). We can identify a one dimensional tiling as a bi-infinite string over A: T = a i 2 a i 1 a i0 a i1 a i2 Language L(T ) is the set of finite substrings of T L(T ) is factorial closed under substrings A marked pattern is now a word in L(T ) with distinguished inand out- letters à and b

10 The free monogenic inverse monoid denote by FIM 1 tiling semigroup of 1-diml vanilla tiling (with 0 removed) elements are marked strings on a single letter t multiply by matching out- and in- markers and superposing strings strings always match, and ť = 1 gap between in- and out- letter is a hom to Z Give coords in Z 3 : t p t p+1 t 1 `t 0 t 1 t r t q (p, q, r)

11 A presentation for S(T ) Let T be a one-dimensional tiling with alphabet A = {a 1,..., a n }. Theorem (McAlister-Soares 2006) S(T ) is generated by the single-tile idempotents ǎ i and the two-tile patterns t ij = à i á j, with defining relations ǎ i 2 = ǎ i, ǎ i ǎ j = 0, ǎ i t ij = t ij = t ij ǎ j t 1 ij t ik = 0 if j k, t ij t 1 kj = 0 if i k w = 0 if w has underlying word L(T )

12 Finite presentability Corollary S(T ) is a finitely presented inverse semigroup iff L(T ) is a locally testable language. Locally testable: determine membership by substrings of fixed length. Factorial and locally testable iff membership determined by finitely many forbidden subwords. More on finite presentability in dimension > 1 later.

13 F inverse 1-diml tiling semigroups are F inverse: each non-zero element is below a unique maximal element in the natural partial order: u û where û is the smallest substring of π carrying the in- and out- tiles of the marked string u. n-diml tiling semigroups (n > 1) need not be F inverse but more of this later.

14 Periodic tilings For detailed structure, simplify to periodic T : repeat fixed finite string ad bi-infinitum... Theorem (Dombi-NDG 2009) abcababcababcababcab The tiling semigroup of a one-dimensional periodic tiling with period of length m embeds into P(Z m ) 0 FIM 1, and the subsemigroups that arise are completely determined.

15 Embeds where? P(Z m ) power set of subsets of Z m FIM 1 free monogenic inverse monoid FIM 1 Z Z m so FIM 1 acts on P(Z m ) by translation P(Z m ) 0 FIM 1 is the semidirect product of monoids, with (, w) = 0 for all w FIM 1.

16 Sketch proof of the theorem T periodic with alphabet A = {t 1, t 2,..., t n } and period p 0 p 1 p m 1. Let π be a marked string in T of length l. Find coords τ(π) Z 3 just by counting (ignore spelling!) If l m, the in-letter is a unique p j and we set Ω(π) = {j}, but if l < m then the in-letter may be p j with j in some finite subset Ω(π). Theorem S(T ) embedded in P(Z m ) 0 FIM 1 by π (Ω(π), τ(π)).

17 Hypercubic tilings of R n McAlister-Soares 2009 tiling R n by (coloured) regular cubic lattice dual graph is the Cayley graph of Z n with coloured vertices tiling is replaced by colouring map χ : Z n Σ

18 Commutators are idempotents: In any tiling semigroup S(T ), u 1 v 1 uv is an idempotent: distance between in-tile and out-tile must be 0.

19 High dimensional, but vanilla For the n diml vanilla hypercubic tiling V: all patterns have non-zero product, so 0 is removable Set M = Inv a, b : (u 1 v 1 uv) 2 = u 1 v 1 uv(u, v {a ±, b ± } + Margolis-Meakin (1989): M is the universal E unitary inverse monoid with max group image Z 2 So have φ : M S(V) Theorem (McAlister-Soares 2009) ker φ is not finitely generated as a congruence on M.

20 LF Y-HNN Locally full Yamamura HNN-extensions of semilattices with zero: E a semilattice, 0 E: (e) = {x E : x e}, Family of isomorphisms φ i : (e i ) (f i ), S = Inv 0 E, t i : t i t 1 i Theorem (Yamamura 1999) = e i, t 1 i t i = f i, t 1 i xt i = xφ i (x e i ) E(S) = E, every subgroup of S is free, and S is (strongly) F inverse.

21 The path extension For an n-diml tiling T, non-zero elements of the path extension Π(T ) are pairs (P, u) where P S(T ) is a marked pattern, u is a reduced path in the dual graph of T from the in-tile to the out-tile of P Multiply componentwise: (P, u)(q, v) = (PQ, uv), where uv is the free reduction of the concatenation of u and v (ie we work in the fundamental groupoid of the dual graph).

22 Multiplication in Π(T )

23 Path extension covers... For any n-diml tiling T : Theorem (Dombi-NDG 2010?) The path extension is a strongly F inverse cover of S(T ) and is isomorphic to a locally full Yamamura HNN extension of E(S(T )). (If n = 1 the covering map is an isomorphism.)

24 Y-HNN ingredients E = E(S(T )) = E(Π(T )), Idempotents e i are all possible two-tile patterns with a fixed choice of marked tile, Idempotents f i choose the other tile, Isom φ : E i F i switches the marking.