Finitely presented nil-semigroups
and aperiodic tilings
SubTile 2013, CIRM, France
Alexey Kanel-Belov, Ilya Ivanov-Pogodaev
Semigroups
Simple definitions
Semigroup: set with an associative binary operation.
Generators or letters: set of semigroup elements such
that all other elements can be presented by them.
Zero: 0 is a nil element iff 0x=x0=0 for any element x
from a semigroup.
We can define the semigroup as the set of finite words in
some alphabet L. This is a free semigroup. But we can
assign some relations between the letters of L, such that
Word1=Word2 or Word=0. These equalities are called
as defining relations.
Examples

set of finite words in a finite alphabet,

positive integers with addition,

ideal of a ring, etc
Example
semigroup of words in the alphabet {a,b}
with relations {a2=0, ba=0}. Non nil
elements are: a, bn, abn for any positive
integer n.
Finitelely presented semigroup: a semigroup with a finite
set of defining relations.
Example of infinite nil semigroup
Nil element: an element x such that xn=0 for some non
negative integer n.
Nil semigroup: a semigroup is nil iff every element is nil.
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semigroup of words in the alphabet {a,b,c}
with infinite set of relations {X2=0} (for any
finite word X).
There are infinite number of non zero
words.
Burnside problem in groups
Groups: element x is periodic of period n if xn=1 for some positive integer n. Group is periodic of
period n iff for any x in G: xn=1.
Burnside Problem
Let G be a finitely generated group of period n. Is it finite?
It is true for n=1,2,3,4,6; counter examples are known: for all odd numbers greater than 660 (P.Novikov
S. Adyan); even numbers : greater than several thousands (S.Ivanov, A.Lyseonok). Geometrical proof of
this based on Van Campen diagram mosaics (can be found in papers of M.Gromov).
Connections between group relations and tilings investigated by J.Conway in the set of papers.
Remark. Nothing is known for n=5!
Finitely presented Burnside
Conjecture
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Is there exist an infinite finitely presented periodic group?
Finitely presented semigroups
Questions
In order to control defining relations in object one have to consider finitely presented objects. By
other words, these objects can be presented by the finite information set.
Some interesting groups, rings of semigroups can be constructed using finite number of defining
relations. There are some questions here. In example V.N.Latyshev set a following problem:
V.N.Latyshev
Is there exist an infinite finitely presented nil-ring?
This talk is devoted to the question of L.N. Shevrin.
L.N Shevrin
Is there exist an infinite finitely presented nil-semigroup?
The main point of this problem
to obtain an global property (Nil-semigroup)
with local tools (finite number of defining
relations).
standard question for tilings
Construct a nonperiodic tiling with a
finite set of tiles
So there are some connections between these topics
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Finitely presented semigroups
Monomial case
Example
There are no infinite finitely presented nil-semigroup, which defining
relations are monomial.
Equivalent description
Let {vi} be the finite set of forbidden words. Suppose that there exists
an infinite word without forbidden subwords. Then, there exists a
periodic infinite word without forbidden subwords.
Mathematical fantasy
If we can multiply not just left-right but also upside-downside then we
can use aperiodic tiling constructions.
Our goal is the realization of this fantasy
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Semigroups and tilings
Three languages
We want to «read» tilings as the words. So the paths on tiling would be words in the semigroups.
Let us set up some connected points between semigroups and tilings.
Semigroup
ab=0
Tiling
monomial relation
abcab
prohibiting boundary condition
word, element of semigroup
Path on mosaic (tiling)
In order to construct nil-semigroup, we want to construct the set of defining relations allowing us to
convert any periodic word to zero. In parralel world of tilings we can use the language of boundary
conditions to obtain the property such that only non periodic tiling would be possible.
possibility to convert any
periodic word to zero
possibility to interdict
periodic tilings
possibility to interdict
periodic tilings
using
using
using
defining relations
boundary conditions
Hierarchic tiles
Then, we can use hierarchic methods for tile constructing to guarantee the absence of periodic tilings.
This can be obtain by theorem of Goodman-Strauss type.
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Language of paths
The most useful approach is the language of paths.
Semigroup
a
Tiling
letter in the semigroup
abcaba
word, element of the
semigroup
abdc=0
monomial relation
abcd=aebd
equal words
Type of nodal point in the mosaic
Path on mosaic (tiling), the
sequence of nodal points
Impossible path
Equivalent paths
This language can be effectively used to set up the various number of tiling conditions and in the
same time can be easily translated to semigroup language.
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Language of paths
Additional ideas
Only aperiodical
tilings are possible
Every «ring» path on the
tiling should be banned
So, every periodic path
should be banned
Good way to fix this: ban all nongeodesic paths
Thus, we construct a hierarchical tiling such that all periodic paths are forbidden or can be converted
to forbidden paths. So should provide a tool for considering which path are periodic.
This tool is the wide possibilities of geodesic path to local changing:
we can change some subpaths several times in every regular path
and locally convert it into another geodesic paths with the same edge
nodes.
The path can «scan» the near part of tiling by local equivalents (rewriting rules)
So, for any two points A and B on the tiling there exist a wide number of
geodesic paths with no «choke points». The number of such geodesic
paths is grow with the distance between A and B
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B
A
Base graph construction
Tile of level 1
This is the main
hieratical cutting. One
tile can be cut to 6
smaller tiles. We are
interesting only in
topology of this tiling.
Properties
every vertice has finite order;
Every path crossing some tile of level
n can be locally converted such that it
cross that tile by its borders;
If the path walk around 2 sides of
some tile it can be locally converted to
the path that walk through another two
sides of that tile
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Tile of level 2
This is second
level of hierarhy.
It is easy to
provide the
following levels.
Some additional technicalities
Restitution of the mosaic’s part
We can mark nodes and edges of the
tiling to provide the tool of restitution.
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Some additional technicalities
Additional gluing
In order to obtain the absence of choke points we use a
special «additional gluing» attached to the base planar
tiling
Base planar tiling
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Thank you for you attention
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