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On a complete set of operations for factorizing codes

Clelia De Felice (2006)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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 complete set of operations for factorizing codes

Clelia De Felice (2010)

RAIRO - Theoretical Informatics and Applications

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 connection of number theory with graph theory

Lawrence Somer, Michal Křížek (2004)

Czechoslovak Mathematical Journal

We assign to each positive integer n a digraph whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a 2 ≡ b ( m o d n ) . 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 duality of submodule lattices

Gábor Czédli, Géza Takách (2000)

Discussiones Mathematicae - General Algebra and Applications

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 semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . 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 ≥ 2 is arbitrary.

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

Let G be an additive finite abelian group. For every positive integer ℓ, let d i s c ℓ ( G ) 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 d i s c ℓ ( G ) for certain finite groups, including cyclic groups, the groups G = C ₂ ⊕ C 2 m 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...

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