The interval inclusion number of a partially ordered set

AbstractA containment representation for a poset P is a map ƒ such that x<y in P if and only if ƒ(x) ⊂ ƒ(y). We introduce the interval inclusion number (or interval number) i(P) as the smallest t such that P has a containment representation f in which each f(x) is the union of at most t intervals. Trivially, i(P)=1 if and only if dim(P)⩽2. Posets with i(P)=2 include the standard n-dimensional poset and all interval orders; i.e. posets of arbitrarily high dimension. In general we have the upper bound i(P) ⩽ ⌈;dim(P)2⌉, with equality holding for the Boolean algebras. For lexicographic composition, i(P) = k and dim(Q) = 2k + 1 imply i(P[Q]) = k + 1. This result and i(B2k) = k imply that testing i(P) ⩽ k for any fixed k ⩾ 2 is NP-complete. Concerning removal theorems, we prove that i(P − x) ⩾ i(P) − 1 when x is a maximal or minimal element of P, and in general i(P − x) ⩾ i(P)2

Generating the algebraic theory of C(X)C(X): the case of partially ordered compact spaces

It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded by $\aleph_1$. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is a $\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms. We also characterise the $\aleph_1$-copresentable partially ordered compact spaces

An approach to basic set theory and logic

The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows for quick proofs of basic set-theoretic identities and logical tautologies, so it is also a good tool to aid one's memory. I do not assume any exposure to euclidean geometry via axioms. Only an experience with transforming algebraic identities is required. The idea is to get students to do proofs right from the get-go. In particular, I avoid entangling students in nuances of logic early on. Basic facts of logic are derived from set theory, not the other way around.Comment: 22 page

Signs in the cd-index of Eulerian partially ordered sets

A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on cd-coefficients (except for the coefficient of c^n)

Compactness of the space of causal curves

We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.Comment: 15 page