‎In this paper we take $mathcal A$ to be the category {bf Pos-S}‎ ‎of $S$-posets‎, ‎for a posemigroup $S$‎, ${mathcal M}_{pd}$ to‎ ‎be the class of partially ordered sequantially-dense‎ ‎monomorphisms and study the categorical properties‎, ‎such as‎ ‎limits and colimits‎, ‎of this class‎. ‎These properties are usually‎ ‎needed to study the homological notions‎, ‎such as injectivity‎, ‎of‎ ‎$S$-...

Let $${\cal M}$$ be an n-cluster tilting subcategory of mod-Λ, where Λ is Artin algebra. S}({\cal M})$$ denote the full S}(\Lambda )$$ , submodule category Λ, consisting all monomorphisms in . We construct two functors from to $$\bmod - \underline {\cal M} $$ finitely presented additive contravariant on stable show that these are full, dense and objective hence provide equivalences between quot...

Take for instance a k-algebra 4 over a field k. Then the usual k-dual DM=Homk(M, k) of a finite dimensional 4-module M satisfies all three properties. However for infinite dimensional 4-modules the condition (D1) does not hold, and for most rings there is even on the level of the finite length modules no duality functor between right and left modules available. The aim of this note is to descri...

Let K be a perfect field of characteristic p > 0, A1 := K〈x, ∂ | ∂x−x∂ = 1〉 be the first Weyl algebra and Z := K[X := x, Y := ∂] be its centre. It is proved that (i) the restriction map res : AutK(A1) → AutK(Z), σ 7→ σ|Z , is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z) | J (τ) = 1} where J (τ) is the Jacobian of τ (note that AutK(Z) = K ∗ ⋉ Γ and if K is not perfect then im(res) 6= Γ); (ii)...

The case for a recent expansion of modern humans from Africa has been strengthened by the finding of monomorphism in part of a Y-linked gene, consistent with the low variability seen in human mitochondrial DNAs.

‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $S$-posets‎, ‎posets with a monotone action of a‎ pomonoid $S$ on them‎, ‎is studied‎. ‎We give a criterion‎, ‎like the Baer condition for injectivity of modules‎, ‎or Skornjakov criterion for injectivity of $S$-sets‎, ‎for the order dense injectivity‎. ‎Also‎, ‎we consider such injectivit...

Let   a family of monomorphisms in the category A ‎To study mathematical notions in a category  A‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎with respect to a class M of its (mono)morphisms‎, ‎one should know some of the categorical properties of the pair (A ‎,M ‎)‎. ‎In this paper we take A to be the‎‎category  Pos-S  of S-partially ordered sets  and   to be a particular‎ class o...