Shadowing for families of endomorphisms of generalized group shifts

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula> be a countable monoid and let <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> be an Artinian group (resp. an Artinian module). Let <inline-formula><tex-math id="M3">\begin{document}$ \Sigma \subset A^G $\end{document}</tex-math></inline-formula> be a closed subshift which is also a subgroup (resp. a submodule) of <inline-formula><tex-math id="M4">\begin{document}$ A^G $\end{document}</tex-math></inline-formula>. Suppose that <inline-formula><tex-math id="M5">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> is a finitely generated monoid consisting of pairwise commuting cellular automata <inline-formula><tex-math id="M6">\begin{document}$ \Sigma \to \Sigma $\end{document}</tex-math></inline-formula> that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of <inline-formula><tex-math id="M7">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M8">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.</p>

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Generating infinite monoids of cellular automata

Journal of Algebra and Its Applications ◽

10.1142/s0219498822502152 ◽

2021 ◽

pp. 2250215

Author(s):

Alonso Castillo-Ramirez

Keyword(s):

Cellular Automata ◽

Normal Subgroups ◽

Finitely Generated ◽

Residually Finite ◽

Residually Finite Group ◽

Group Of Units ◽

Finite Dimensional ◽

Index Condition ◽

Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] 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 [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

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Duality for dyadic triangles

International Journal of Algebra and Computation ◽

10.1142/s0218196718500637 ◽

2019 ◽

Vol 29 (01) ◽

pp. 61-83 ◽

Cited By ~ 1

Author(s):

K. Matczak ◽

A. Mućka ◽

A. B. Romanowska

Keyword(s):

Binary Operation ◽

Arithmetic Mean ◽

Unit Cube ◽

Unit Interval ◽

Real Plane ◽

Finitely Generated ◽

One Dimensional ◽

Additional Constant ◽

The One ◽

Dyadic Intervals

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

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CELLULAR AUTOMATA OVER SEMI-DIRECT PRODUCT GROUPS: REDUCTION AND INVERTIBILITY RESULTS

International Journal of Algebra and Computation ◽

10.1142/s0218196706003402 ◽

2006 ◽

Vol 16 (06) ◽

pp. 1071-1085

Author(s):

SILVIO CAPOBIANCO

Keyword(s):

Cellular Automata ◽

Direct Product ◽

Configuration Spaces ◽

Finitely Generated ◽

Finitely Generated Groups ◽

Current State ◽

Global Properties ◽

Finite Factor ◽

Finiteness Properties

Cellular automata are transformations of configuration spaces over finitely generated groups, such that the next state in a point only depends on the current state of a finite neighborhood of the point itself. Many questions arise about retrieving global properties from such local descriptions, and finding algorithms to perform these tasks. We consider the case when the group is a semi-direct product of two finitely generated groups, and show that a finite factor (whatever it is) can be thought of as part of the alphabet instead of the group, preserving both the dynamics and some "finiteness" properties. We also show that, under reasonable hypotheses, this reduction is computable: this leads to some reduction theorems related to the invertibility problem.

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Artinian modules over commutative rings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075125 ◽

1992 ◽

Vol 111 (1) ◽

pp. 25-33 ◽

Cited By ~ 11

Author(s):

R. Y. Sharp

Keyword(s):

Commutative Ring ◽

Local Ring ◽

Commutative Rings ◽

Finitely Generated ◽

Finitely Generated Module ◽

Artinian Modules ◽

Noetherian Local Ring ◽

Matlis Duality ◽

Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

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DYADIC POLYGONS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006248 ◽

2011 ◽

Vol 21 (03) ◽

pp. 387-408 ◽

Cited By ~ 7

Author(s):

K. MATCZAK ◽

A. B. ROMANOWSKA ◽

J. D. H. SMITH

Keyword(s):

Binary Operation ◽

Arithmetic Mean ◽

Real Plane ◽

Finitely Generated ◽

Idempotent Algebra ◽

One Dimensional ◽

The Real ◽

Pythagoras Theorem ◽

The One ◽

Dyadic Intervals

Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry.

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Shadowing Property on Finitely Generated Group Actions

Journal of Dynamical Systems and Geometric Theories ◽

10.1080/1726037x.2014.922253 ◽

2014 ◽

Vol 12 (1) ◽

pp. 69-79

Author(s):

Ali Barzanouni

Keyword(s):

Group Actions ◽

Finitely Generated ◽

Shadowing Property ◽

Finitely Generated Group

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SHADOWING ON FRACTALS

Fractals ◽

10.1142/s0218348x94000387 ◽

1994 ◽

Vol 02 (02) ◽

pp. 307-310

Author(s):

FERNANDA BOTELHO ◽

MAX GARZON

Keyword(s):

Neural Networks ◽

Computer Simulation ◽

Dynamical Systems ◽

Cellular Automata ◽

Cantor Sets ◽

Simulation And Modeling ◽

Small Perturbations ◽

Shadowing Property ◽

Finite Approximations ◽

Approximation Errors

We consider several families of continous dynamical systems on Cantor sets arising, in particular, from computer simulation and modeling of neural networks by discrete and/or finite approximations (such as cellular automata). It is shown that such approximations are always observable (have the shadowing property), in the sense that pseudo-orbits obtained by small perturbations of an orbit are approximated by actual orbits. It follows that the true behavior of locally defined dynamical systems can be observed exactly on computer simulations, despite unavoidable discretization and approximation errors.

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On surjunctive monoids

International Journal of Algebra and Computation ◽

10.1142/s0218196715500113 ◽

2015 ◽

Vol 25 (04) ◽

pp. 567-606 ◽

Cited By ~ 3

Author(s):

Tullio Ceccherini-Silberstein ◽

Michel Coornaert

Keyword(s):

Cellular Automata ◽

The Other ◽

Finite Alphabet ◽

Finitely Generated ◽

Commutative Monoids ◽

Residually Finite ◽

Finite Monoids ◽

Other Hand ◽

Bicyclic Monoid

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M 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.

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