We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of B3 defines a left ordering of the free group on k generators, k > 1, that has no convex subgroups. A group G is said to be left-orderable if there exists a strict total ordering of its elements such that g < h implies fg < fh for all f, g, h in G. To each left ordering < of a group G, we can associate the s...

Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend the main result of [9] to the case of uncountable groups. With minor technical restrictions on the group G, we find that no dense left ordering is isolated in LO(G), and that the closu...

Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linear programming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The r...

We study the orbits of G = GL(V ) in the enhanced nilpotent cone V ×N , where N is the variety of nilpotent endomorphisms of V . These orbits are parametrized by bipartitions of n = dimV , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition a...

We use the idea of a scalarization of a cone metric to prove that the topology generated by any cone metric is equivalent to a topology generated by a related metric. We then analyze the case of an ordering cone with empty interior and we provide alternative definitions based on the notion of interior cone. Finally we discuss the implications of such cone metrics in the theory of iterated funct...

S. Sherman has shown [4] that if the self adjoint elements of a C* algebra form a lattice under their natural ordering the algebra is necessarily commutative. In this note we extend this result to real Banach algebras with an identity and arbitrary Banach * algebras with an identity. The central fact for a real Banach algebra A is that if the positive cone is defined to be the uniform closure o...

Miranda-Thompson majorization is a group-induced cone ordering on $mathbb{R}^{n}$ induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize Miranda-Thompson majorization on the matrices. For $X$, $Yin M_{m,n}$, $X$ is said to be  Miranda-Thompson majorized by $Y$ (denoted by $Xprec_{mt}Y$) if there exists some $Din rm{Conv(G)}$ s...

In this paper, we consider an approximation of the ordering cone by a sequence of convex well-based cones that are contained inside the original cone. We derive related results on approximating minimal point sequences and propose a density result being evocative of the well-known ABB theorem.

Efficient points are obtained for cone-ordered maximizations in n R using the method of scalarization. Various scalarizations are presented for ordering cones in general and then for the important special case of polyhedral cones. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented.

In this paper, we present the notion of multimodular triangulation under a new geometrical point of view. We also show the link with multimodular functions by a new proof of the convexity theorem. This is used to de ne a partial ordering compatible with multimodularity called the cone ordering. An application in admission control in queues is then presented.