### A Bruhat order for the class of $(0,1)$-matrices with row sum vector $R$ and column sum vector $S$.

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For every countable ordinal α, we construct an ${l}_{1}$-predual ${X}_{\alpha}$ which is isometric to a subspace of $C\left({\omega}^{{\omega}^{{\omega}^{\alpha}+2}}\right)$ and isomorphic to a quotient of $C\left({\omega}^{\omega}\right)$. However, ${X}_{\alpha}$ is not isomorphic to a subspace of $C\left({\omega}^{{\omega}^{\alpha}}\right)$.

A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

In [3] a metric on a system of isomorphism classes of ordered sets was defined. In this paper we define another metric on the same system and investigate some of its properties. Our approach is motivated by a problem from practice.

In this paper, we establish a theorem on Möbius inversion over power set lattices which strongly generalizes an early result of Whitney on graph colouring.

Let $\mathbb{G}$ and $\mathbb{D}$, respectively, denote the partially ordered sets of homomorphism classes of finite undirected and directed graphs, respectively, both ordered by the homomorphism relation. Order theoretic properties of both have been studied extensively, and have interesting connections to familiar graph properties and parameters. In particular, the notion of a duality is closely related to the idea of splitting a maximal antichain. We construct both splitting and non-splitting infinite maximal antichains...

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...

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_{c}$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays...