338 research outputs found
On surjunctive monoids
A monoid is called surjunctive if every injective cellular automata with
finite alphabet over is surjective. We show that all finite monoids, all
finitely generated commutative monoids, all cancellative commutative monoids,
all residually finite monoids, all finitely generated linear monoids, and all
cancellative one-sided amenable monoids are surjunctive. We also prove that
every limit of marked surjunctive monoids is itself surjunctive. On the other
hand, we show that the bicyclic monoid and, more generally, all monoids
containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive
Von Neumann Regular Cellular Automata
For any group and any set , a cellular automaton (CA) is a
transformation of the configuration space defined via a finite memory set
and a local function. Let be the monoid of all CA over .
In this paper, we investigate a generalisation of the inverse of a CA from the
semigroup-theoretic perspective. An element is von
Neumann regular (or simply regular) if there exists
such that and , where is the composition of functions. Such an
element is called a generalised inverse of . The monoid
itself is regular if all its elements are regular. We
establish that is regular if and only if
or , and we characterise all regular elements in
when and are both finite. Furthermore, we study
regular linear CA when is a vector space over a field ; in
particular, we show that every regular linear CA is invertible when is
torsion-free elementary amenable (e.g. when ) and , and that every linear CA is regular when
is finite-dimensional and is locally finite with for all .Comment: 10 pages. Theorem 5 corrected from previous versions, in A.
Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata
and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer,
201
Inverse monoids and immersions of 2-complexes
It is well known that under mild conditions on a connected topological space
, connected covers of may be classified via conjugacy
classes of subgroups of the fundamental group of . In this paper,
we extend these results to the study of immersions into 2-dimensional
CW-complexes. An immersion between
CW-complexes is a cellular map such that each point has a
neighborhood that is mapped homeomorphically onto by . In order
to classify immersions into a 2-dimensional CW-complex , we need to
replace the fundamental group of by an appropriate inverse monoid.
We show how conjugacy classes of the closed inverse submonoids of this inverse
monoid may be used to classify connected immersions into the complex
Generating infinite monoids of cellular automata
For a group and a set , let be the monoid of all
cellular automata over , and let be its group of units.
By establishing a characterisation of surjunctuve groups in terms of the monoid
, we prove that the rank of (i.e. the
smallest cardinality of a generating set) is equal to the rank of
plus the relative rank of in
, and that the latter is infinite when has an infinite
decreasing chain of normal subgroups of finite index, condition which is
satisfied, for example, for any infinite residually finite group. Moreover,
when is a vector space over a field , we study the monoid
of all linear cellular automata over and
its group of units . We show that if is an
indicable group and is finite-dimensional, then
is not finitely generated; however, for any
finitely generated indicable group , the group
is finitely generated if and only if
is finite.Comment: 11 page
On Factor Universality in Symbolic Spaces
The study of factoring relations between subshifts or cellular automata is
central in symbolic dynamics. Besides, a notion of intrinsic universality for
cellular automata based on an operation of rescaling is receiving more and more
attention in the literature. In this paper, we propose to study the factoring
relation up to rescalings, and ask for the existence of universal objects for
that simulation relation. In classical simulations of a system S by a system T,
the simulation takes place on a specific subset of configurations of T
depending on S (this is the case for intrinsic universality). Our setting,
however, asks for every configurations of T to have a meaningful interpretation
in S. Despite this strong requirement, we show that there exists a cellular
automaton able to simulate any other in a large class containing arbitrarily
complex ones. We also consider the case of subshifts and, using arguments from
recursion theory, we give negative results about the existence of universal
objects in some classes
On Residually Finite Semigroups of Cellullar Automata
We prove that if is a monoid and a finite set with more than one
element, then the residual finiteness of is equivalent to that of the
monoid consisting of all cellular automata over with alphabet
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