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We characterize the absolute retracts in the category of reflexive oriented graphs, that is, antisymmetric reflexive graphs, where morphisms between objects preserve arcs (which may be sent to loops). Here we show, by correcting a much earlier attempt at a proof, that a reflexive oriented graph is an absolute retract if and only if it is indeed a retract of some (direct) product of reflexive oriented zigzags (which are concatenations of reflexive oriented paths). Absolute retracts are therefore necessarily acyclic. In contrast to other categories of graphs and ordered sets, not every acyclic oriented graph can be embedded isometrically into some absolute retract. Embedding involves isometry with respect to the zig-zag distances forming a particular "dual quantale", which is a complete lattice of certain sets of words over the alphabet $\{+, -\}$, endowed with an additional monoid operation (viz., compound concatenation of sets of words) and an involution (interchanging $+$ and $- $ and then mirroring words). As reflexive oriented zigzags have MacNeille-closed distances, so do their products and retracts. So, the category of reflexive oriented graphs and its full subcategory of reflexive acyclic graphs do not have enough injectives, as the injective objects coincide with the absolute retracts., Comment: 6 figures

Let $\Lambda^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $\Lambda$, which is equipped with some (partial) order. This ordering lifts to $\Lambda^{\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $\Lambda^{\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in $\Lambda^*$ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $\Lambda$, one can easily identify the closed lower sets as the finitely generated ones, whereas it is more complicated to determine the closed upper sets. For a fairly large class of ordered sets $\Lambda$ (including complete lattices as well as antichains) one can generate the closure of any upper set of words by means of binary operations ( "syntactic rules") thus obtaining an efficient procedure to test closedness. Closed upper set of words are involved in an embedding theorem for valuated oriented graphs. In fact, generalized paths (so-called "zigzags") are encoded by words over an alphabet $\Lambda$. Then the valuated oriented graphs which are "isometrically" embeddable in a product of zigzags have the characteristic property that the words corresponding to the zigzags between any pair of vertices form a closed upper set in $\Lambda$., Comment: 18pages, 3 figures

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs., Comment: 40 pages, 6 figures, (improved exposition)

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon., Comment: 27 pages, 6 figures