It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set $𝒪$ of operations exists such that each factorizing code can be obtained by using the operations in $𝒪$ and starting with prefix or suffix codes. $𝒪$ is named here a complete set of operations (for factorizing codes). We show...

It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set O of operations exists such that each factorizing code can be obtained by using the operations in O and starting with prefix or suffix codes. O is named here a complete set of operations (for factorizing codes). We show...

We assign to each positive integer $n$ a digraph whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\left(modn\right)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

A $B\left(2\right)$-group is a sum of a finite number of torsionfree Abelian groups of rank $1$, subject to two independent linear relations. We complete here the study of direct decompositions over two base elements, determining the cases where the relations play an essential role.

Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...

It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable, then $A$ is either $p^{\omega +n}$-totally projective or $p^{\omega +n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega +n}$-totally projective or strongly $p^{\omega +n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch....