We prove that the multiplicity of a simple module as a composition factor in a composition series for a primitive band module over a domestic string algebra is at most two.

We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor ${ℍ}_C:C-Comod\to H_C-Comod$ that restricts to a representation equivalence ${ℍ}_C:C-comod\to H_C-como{d}_{sp}^{•}$, where $H_C$ is a coradical square complete hereditary bipartite...

A class of stratified posets $I{*}_{\varrho }$ is investigated and their incidence algebras $KI{*}_{\varrho }$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I{*}_{\varrho }$ we associate with $I{*}_{\varrho }$ a bound quiver (Q, Ω) in such a way that $KI{*}_{\varrho }\simeq K(Q,\Omega )$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I{*}_{\varrho }$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_{\varrho }$ is three-partite a fundamental domain $I^{*+\times }$ of this covering is...