On surjunctive monoids

Petri nets with generalized algebra: a comparison

В наших предыдущих работах мы продемонстрировали, что теория конечных подмножеств различных ассоциативных алгебр позволяет интерпретировать элементарную арифметику, в частности, она неразрешима. Например, это было показано для любых бесконечных абелевых групп. Возникает естественный вопрос: можно ли обобщить этот результат на более широкий класс алгебр, скажем, все коммутативные моноиды. В некоторых случаях нами ответ тоже получен ранее: для коммутативных моноидов с сокращением, имеющим элемент бесконечного порядка или произвольных абелевых групп. Сейчас же мы продемонстрируем, что не для всяких коммутативных моноидов это верно. Более того, мы дадим описание конструкции, которая позволяет строить такого рода системы из разного рода исходных алгебр. Вместе с тем, будут указаны и некоторые границы её применимости. In our previous works, we have proved for various associative algebras that the finite subsets theory allows to interpret elementary arithmetic, in particular, such theory is undecidable. For example, this is proved for all infinite Abelian groups. A natural question arises: can we generalize this result to a wider class of algebras, for example, all commutative monoids. In some cases, we also have proved analogous result: for commutative cancellative monoids with an element of infinite order, or arbitrary Abelian groups. In this paper we prove that this is not true for arbitrary commutative monoids. Moreover, we propose a method that allows to construct such algebras by various original algebras. Also, we have found a limitation of this method.

A generalization of the Schützenberger product of finite monoids

Let H be a cancellative commutative monoid, let mathcal {A}(H) be the set of atoms of H and let widetilde{H} be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counterexamples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if widetilde{H} is a DVM, then H is transfer Krull if and only if Hsubseteq widetilde{H} is inert. Moreover, we prove that if widetilde{H} is factorial, then H is transfer Krull if and only if mathcal {A}(widetilde{H})={uvarepsilon mid uin mathcal {A}(H),varepsilon in widetilde{H}^{times }}. We also show that if widetilde{H} is half-factorial, then H is transfer Krull if and only if mathcal {A}(H)subseteq mathcal {A}(widetilde{H}). Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.

For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of groups that depends on the numbers $\alpha_{[H]}$ of periodic configurations for conjugacy classes $[H]$ of subgroups of $G$. We show how the numbers $\alpha_{[H]}$ may be computed using the M\"obius function of the subgroup lattice of $G$, and we use this to improve the lower bound recently found by Gao, Jackson and Seward on the number of aperiodic configurations of $A^G$. Furthermore, we study generating sets of $\text{CA}(G;A)$; in particular, we prove that $\text{CA}(G;A)$ cannot be generated by cellular automata with small memory set, and, when all subgroups of $G$ are normal, we determine the relative rank of $\text{ICA}(G;A)$ on $\text{CA}(G;A)$, i.e. the minimal size of a set $V \subseteq \text{CA}(G;A)$ such that $\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$.

Our main result is the following: let $X$ be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface $X^{\prime}$ with automorphism group isomorphic to the automorphism group of $X$ if and only if $X$ is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety $X$ of any dimension. Recall that normalized additive group actions on $X$ are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of $X$. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

Early we (with B. N. Karlov) have proved the following claim for the infinite cyclic monoid ℋ. Let exp ℋ be an algebra of finite subsets of ℋ with the same operation, exp ℋ must be a monoid again. So the theory of exp ℋ is equivalent to elementary arithmetic. Thus, the theory of the monoid exp ℋ is undecidable. Here we consider an arbitrary commutative cancellative monoid ℋ with an element of infinite order, and generalize the previous claims to the corresponding monoid exp ℋ.

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

A property of superposability is defined here for Treillis Automata as an equivalent of the linearity property of Cellular Automata. We take advantage of the binary local transition function of TAs, which enables us to use an infix notation for it, to study this property in an algebraic framework. We show that a TA is superposable iff its local transition function operates on its set of states as a commutative monoid law and that the general case of commutative monoids is reducible to the case of Abelian groups in which the space-time diagrams obtained are isomorphic to a superposition of figures representing the triangle of binomial coefficients (Pascal's triangle) modulo integers.KeywordsCellular AutomatonInitial ConfigurationNeutral ElementBinomial CoefficientCommutative MonoidThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider left permutive cellular automata $\Phi$ with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points $x$ such that $\Phi : \operatorname {cl} \{\Phi ^n(x) : n \ge 0\} \to \operatorname {cl} \{\Phi ^n(x) : n \ge 0\}$ is topologically conjugate to an odometer, the “$+1$” map on the countable product of finite cyclic groups. This set is a dense $G_\delta$ subset of an appropriate subspace. We identify the odometer in several cases.

Theimplicationsofa specific pseudometricon the collection of languages over a finite alphabet are explored. In distinction from an approach in (Calude et al., 2009) that relates to collections of infinite or bi-infinite sequences, the present work is based on an adaptation of the “Besicovitch” pseudometric introduced by Besicovitch (1932) and elaborated in (Cattaneo et al., 1997) in the context of cellular automata. Using this pseudometric to form a metric quotient space, we study its propertiesanddrawconclusionsaboutthe locationof certain well-understood families of languages in the language space. We find that topologies, both on the space of formal languages itself and upon quotient spaces derived from pseudometrics on the language space, may offer insights into the relationships, and in particular the distance, between languages over a common alphabet.

Markovianity of the invariant distribution of probabilistic cellular automata on the line

Complemented MacNeille completions and algebras of fractions