10 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
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 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
On sofic monoids
We investigate the notion of soficity for monoids. A group is sofic as a
group if and only if it is sofic as a monoid. All finite monoids, all
commutative monoids, all free monoids, all cancellative one-sided amenable
monoids, all multiplicative monoids of matrices over a field, and all monoids
obtained by adjoining an identity element to a semigroup without identity
element are sofic. On the other hand, although the question of the existence of
a non-sofic group remains open, we prove that the bicyclic monoid is not sofic.
This shows that there exist finitely presented amenable inverse monoids that
are non-sofic.Comment: We have corrected a small mistake in (and then suitably refrmulated)
the statement of Theorem 6.1 (we needed the "Left-cancellative" hypothesis on
M. It will appear in SEMIGRUOP FORU
The algebraic entropy of one-dimensional finitary linear cellular automata
The aim of this paper is to present one-dimensional finitary linear cellular
automata on from an algebraic point of view. Among various
other results, we:
(i) show that the Pontryagin dual of is a classical
one-dimensional linear cellular automaton on ;
(ii) give several equivalent conditions for to be invertible with inverse
a finitary linear cellular automaton;
(iii) compute the algebraic entropy of , which coincides with the
topological entropy of by the so-called Bridge Theorem.
In order to better understand and describe the entropy we introduce the
degree and of and .Comment: 21 page
Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture
Sofic and hyperlinear groups are the countable discrete groups that can be
approximated in a suitable sense by finite symmetric groups and groups of
unitary matrices. These notions turned out to be very deep and fruitful, and
stimulated in the last 15 years an impressive amount of research touching
several seemingly distant areas of mathematics including geometric group
theory, operator algebras, dynamical systems, graph theory, and more recently
even quantum information theory. Several longstanding conjectures that are
still open for arbitrary groups were settled in the case of sofic or
hyperlinear groups. These achievements aroused the interest of an increasing
number of researchers into some fundamental questions about the nature of these
approximation properties. Many of such problems are to this day still open such
as, outstandingly: Is there any countable discrete group that is not sofic or
hyperlinear? A similar pattern can be found in the study of II_1 factors. In
this case the famous conjecture due to Connes (commonly known as the Connes
embedding conjecture) that any II_1 factor can be approximated in a suitable
sense by matrix algebras inspired several breakthroughs in the understanding of
II_1 factors, and stands out today as one of the major open problems in the
field. The aim of these notes is to present in a uniform and accessible way
some cornerstone results in the study of sofic and hyperlinear groups and the
Connes embedding conjecture. The presentation is nonetheless self contained and
accessible to any student or researcher with a graduate level mathematical
background. An appendix by V. Pestov provides a pedagogically new introduction
to the concepts of ultrafilters, ultralimits, and ultraproducts for those
mathematicians who are not familiar with them, and aiming to make these
concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto
Subshifts with Simple Cellular Automata
A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of subshifts are called cellular automata, and we call the set of cellular automata on a subshift its endomorphism monoid. It is known that the set of all sequences (the full shift) allows cellular automata with complex dynamical and computational properties. We are interested in subshifts that do not support such cellular automata. In particular, we study countable subshifts, minimal subshifts and subshifts with additional universal algebraic structure that cellular automata need to respect, and investigate certain criteria of ‘simplicity’ of the endomorphism monoid, for each of them. In the case of countable subshifts, we concentrate on countable sofic shifts, that is, countable subshifts defined by a finite state automaton. We develop some general tools for studying cellular automata on such subshifts, and show that nilpotency and periodicity of cellular automata are decidable properties, and positive expansivity is impossible. Nevertheless, we also prove various undecidability results, by simulating counter machines with cellular automata. We prove that minimal subshifts generated by primitive Pisot substitutions only support virtually cyclic automorphism groups, and give an example of a Toeplitz subshift whose automorphism group is not finitely generated. In the algebraic setting, we study the centralizers of CA, and group and lattice homomorphic CA. In particular, we obtain results about centralizers of symbol permutations and bipermutive CA, and their connections with group structures.Siirretty Doriast