Given a tournament T = (V, A), with each subset X of V is associated the subtournament T [X] = (X, A∩(X×X)) of T induced by X. A subset I of V is an interval of T provided that for any a, b ∈ I and x ∈ V \I, (a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, {x}, where x ∈ V , and V are intervals of T called trivial. A tournament is indecomposable if all its intervals are trivial; otherwise,...

We show that if G is a group and G has a graphproduct decomposition with finitely-generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the...

Beimel, Tassa and Weinreb (2008) and Farras and Padro (2010) partially characterized access structures of ideal weighted threshold secret sharing schemes in terms of the operation of composition. They classified indecomposable ideal weighted threshold access structures, and proved that any other ideal weighted threshold access structure is a composition of indecomposable ones. It remained uncle...

It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable subrings A = A1 ⊕ · · · ⊕ An such that the Ai-Mod are indecomposable module categories. Similarly any coalgebra C over a field can be written as a direct sum of indecomposable subcoalgebras C = ⊕ I Ci such that the categories of Ci-comodules are indecomposable. In this paper a decomposit...

This research began as an effort to determine exactly which one-dimensional local rings have indecomposable finitely generated modules of arbitrarily large constant rank. The approach, which uses a new construction of indecomposable modules via the bimodule structure on certain Ext groups, turned out to be effective mainly for hypersurface singularities. The argument was eventually replaced by ...

A permutation a1a2 . . . an is indecomposable if there does not exist p < n such that a1a2 . . . ap is a permutation of {1, 2, . . . , p}. We compute the asymptotic probability that a permutation of Sn with m cycles is indecomposable as n goes to infinity with m/n fixed. The error term is O( log(n−m) n−m ). The asymptotic probability is monotone in m/n, and there is no threshold phenomenon: it ...