On A Combinatorial Problem Connected withFactorizations
On a complete set of operations for factorizing codes
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 of operations exists such that each factorizing code can be obtained by using the operations in and starting with prefix or suffix codes. is named here a complete set of operations (for factorizing codes). We show...
On a complete set of operations for factorizing codes
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...
On a conjecture of Erdös and Heilbronn
On a conjecture of Hamidoune for subsequence sums.
On a connection of number theory with graph theory
We assign to each positive integer a digraph whose set of vertices is and for which there is a directed edge from to if . 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.
On a problem of A.D. Sands.
On a variation of Sands' method.
On additive bases
On additive bases with two elements
On duality of submodule lattices
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.
On Erdős-Ginzburg-Ziv inverse theorems
On factorizations of finite Abelian groups which admit replacement of a Z-set by a subgroup.
On finite commutative IP-loops with elementary abelian inner mapping groups of order
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order are centrally nilpotent of class at most two.
On invariants related to non-unique factorizations in block monoids and rings of algebraic integers
On multiplicative bases in abelian groups
On restricted sumsets in abelian groups of odd order.
On semiregular digraphs of the congruence
We assign to each pair of positive integers and a digraph whose set of vertices is and for which there is a directed edge from to if . The digraph is semiregular if there exists a positive integer such that each vertex of the digraph has indegree or 0. Generalizing earlier results of the authors for the case in which , we characterize all semiregular digraphs when is arbitrary.
On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths
Let G be an additive finite abelian group. For every positive integer ℓ, let 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 for certain finite groups, including cyclic groups, the groups 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...
On short zero-sum subsequences. II.