A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of ...

A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination relevant comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

Let E be a Frobenius category, let E denote its stable category. The shift functor on E induces a first shift functor on the category of acyclic complexes with entries in E by pointwise application. Shifting a complex by 3 positions yields a second shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that these two shift functors become isomorp...

Let E be a Frobenius category, let E denote its homotopy category. The shift functor on E induces a first shift functor on the category of acyclic complexes with entries in E by pointwise application. Shifting a complex by 3 positions yields a second shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that these two shift functors become isomo...

Let E be a Frobenius category. Let E denote its stable category. The shift functor on E induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in E . Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomor...