An elementary proof is given for Hutchinson's duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order $p^4$ are centrally nilpotent of class at most two.

We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ 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 $a^k\equiv b(modn)$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\ge 2$ is arbitrary.

Let G be an additive finite abelian group. For every positive integer ℓ, let $dis{c}_{\ell }\left(G\right)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $dis{c}_{\ell }\left(G\right)$ for certain finite groups, including cyclic groups, the groups $G=C₂\oplus C_{2m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...

The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class group satisfies...

The investigation of the counting function of the set of integral elements, in an algebraic number field, with factorizations of at most k different lengths gives rise to a combinatorial constant depending only on the class group of the number field and the integer k. In this paper the value of these constants, in case the class group is an elementary p-group, is estimated, and determined under additional conditions. In particular, it is proved that for elementary 2-groups these constants are equivalent...

For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S=g₁·...·g_l$ over G such that $(X^{g₁}-a₁)·...·(X^{g_l}-a_l)\ne 0\in K\left[G\right]$ for all $a₁,...,a_l\in K^{\times }$. If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

A power digraph, denoted by $G(n,k)$, is a directed graph with ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined....