Stone space partitions $$\{X_{p}\mid p\in P\}$$ satisfying conditions like $$\overline{X_{p}}=\bigcup _{q\leqslant p}X_{q}$$ for all $$p\in P$$ , where P is a poset or PO system (poset with distinguished subset), arise naturally in the study both of primitive Boolean algebras and $$\omega $$ -categorical structures. A key concept studying such that p-trim open set which meets precisely those $$...

in this paper the notion of rees short exact sequence for s-posets is introduced, and we investigate the conditions for which these sequences are left or right split. unlike the case for s-acts, being right split does not imply left split. furthermore, we present equivalent conditions of a right s-poset p for the functor hom(p;-) to be exact.

We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbo...

We introduce Priestley rings of upsets (of a poset) and prove that inequivalent Priestley ring representations of a bounded distributive lattice L are in 1-1 correspondence with dense subspaces of the Priestley space of L. This generalizes a 1955 result of Bauer that inequivalent reduced field representations of a Boolean algebra B are in 1-1 correspondence with dense subspaces of the Stone spa...

fuzzy order congruences play an important role in studying the categoricalproperties of fuzzy posets. in this paper, the correspondence between the fuzzyorder congruences and the fuzzy order-preserving maps is discussed. we focus onthe characterization of fuzzy order congruences on the fuzzy poset in terms ofthe fuzzy preorders containing the fuzzy partial order. at last, fuzzy completecongruen...

This article illustrates the dynamical concept of homomesy in three kinds systems – combinatorial, piecewise-linear, and birational shows relationship between these settings. In particular, we show how rowmotion promotion operations Striker Williams [16] can be lifted to (continuous) piecewise-linear on order polytope Stanley [14], then positive orthant ? |P| indeed a dense subset ? . When pose...

We consider the follow poset merging problem: Let X and Y be two subsets of a partially ordered set S. Given complete information about the ordering within both X and Y , how many comparisons are needed to determine the order on X ∪ Y ? This problem is a natural generalization of the list merging problem. While searching in partially ordered sets and sorting partially ordered sets have been con...

In this paper, we introduce the concept of meet precontinuous posets, a generalization of meet continuous lattices to posets. The main results are: (1) A poset P is meet precontinuous iff its normal completion is a meet continuous lattice iff a certain system γ(P ) which is, in the case of complete lattices, the lattice of all Scott-closed sets is a complete Heyting algebra; (2) A poset P is pr...

Intuitively, a set is considered to be discrete if it is composed of isolated elements, whereas it is considered to be continuous if it is composed of infinite and contiguous elements and does not contain “holes”. More formally, to introduce the concept of continuous optimization, some preliminary definitions are required. If we consider sub-sets of real numbers, where the partial order is obvi...

A poset is (r+s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r− 1)(s− 1)w chains, where w is the width of P . This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992–1999, 2010).