Generating infinite monoids of cellular automata

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.

On the group of unit-valued polynomial functions

Let R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let R be a commutative chain ring with invariants p,n,r,k,m. It is known that R is an Eisenstein extension of degree k of a Galois ring S=GR(pn,r). If p−1 does not divide k, the structure of the unit group U(R) is known. The case (p−1)∣k was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of U(R). In this article, we manage to determine the structure of U(R) when (p−1)∣k by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.

QUANTITATIVELY NOMINAL RELATED PHRASES

As can be seen from the title of the paragraph, in this case we will talk about one type of related subordinate phrases, which is formed not in the structure of the sentence, but as a "pre-sentence", an independent unit of nomination. In this group of units, such constructions are represented as quantitative and nominal (ten books, many people, several students, a group of climbers, a herd of horses, etc.), combinations with the meaning of compatibility (Mom and I, brother and sister, etc. .), combinations with the meaning of selectivity (one of us, one of the representatives, etc.), etc.

Finite unitary rings with a single subgroup of prime order of the group of units

Let [Formula: see text] be a unitary ring of finite cardinality [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text]. We show that if the group of units of [Formula: see text] has at most one subgroup of order [Formula: see text], then [Formula: see text] where [Formula: see text] is a finite ring of order [Formula: see text] and [Formula: see text] is a ring of cardinality [Formula: see text] which is one of the six explicitly described types.

Dot product graphs and domination number

Let A be a commutative ring with 1≠0 and R=A×A. The unit dot product graph of R is defined to be the undirected graph UD(R) with the multiplicative group of units in R, denoted by U(R), as its vertex set. Two distinct vertices x and y are adjacent if and only if x·y=0∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when $A=\mathbb {Z}_{n}$ A = ℤ n , $n \in \mathbb {N}$ n ∈ ℕ , n≥2. Inspired by this idea, we study this graph when A has a finite multiplicative group of units. We define the congruence unit dot product graph of R to be the undirected graph CUD(R) with the congruent classes of the relation $\thicksim $ ∽ defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring $R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$ R = ℤ n × ... × ℤ n , k times and k<∞, where all elements of the ring are vertices and adjacency of two distinct vertices is the same as in UD(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .