A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures

Globally simple Heffter arrays and orthogonal cyclic cycle decompositions

In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In particular we provide explicit constructions of such decompositions for cycles of length $k\leq 10$. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.Comment: The present version also considers the problem of biembedding

A generalization of Heffter arrays

In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$. A $H_t(m,n; s,k)$ Heffter array over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in \mathbb{Z}_v\setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e. with entries chosen in $\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by $H_k(n;k)$. In particular, we prove that for $3\leq k\leq n$, with $k\neq 5$, there exists an integer $H_k(n;k)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\pmod 4$; (b) $k\equiv 2\pmod 4$ and $n$ is even; (c) $k\equiv 0\pmod 4$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph

SOME REMARKS ON THE PROBABILITY OF GENERATING AN ALMOST SIMPLE GROUP

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Strongly monogenic a-rigid nearrings

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Composition rings from formal power series rings

We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also