Spectral partially ordered sets

We examine some topics related to (gold)spectral partially ordered sets, i.e., those that are order isomorphic to (Goldman) prime spectra of commutative rings. Among other results, we characterize the partially ordered sets that are isomorphic to prime spectra of rings satisfying the descending chain condition on radical ideals, and we show that a dual of a tree is isomorphic to the Goldman prime spectrum of a ring if and only if every chain has an upper bound. We also give some new methods for constructing (gold)spectral partially ordered sets.     

Component partially ordered sets

Properties of component partially ordered sets (i.e., dense subsets of Boolean algebras) are used to construct mappings of Boolean algebras generalizing the idea of homomorphisms; the properties of a minimal Boolean algebra generated by a given component partially ordered set are investigated.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 275–283, March, 1971.     

Algorithms over partially ordered sets

We here study some problems concerned with the computational analysis of finite partially ordered sets. We begin (in § 1) by showing that the matrix representation of a binary relationR may always be taken in triangular form ifR is a partial ordering. We consider (in § 2) the chain structure in partially ordered sets, answer the combinatorial question of how many maximal chains might exist in a partially ordered set withn elements, and we give an algorithm for enumerating all maximal chains. We give (in § 3) algorithms which decide whether a partially ordered set is a (lower or upper) semi-lattice, and whether a lattice has distributive, modular, and Boolean properties. Finally (in § 4) we give Algol realizations of the various algorithms.     

Arithmetic progressions in partially ordered sets

Van der Waerden's arithmetic sequence theorem—in particular, the density version of Szemerédi—is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and >0. Then for every sufficiently large partially ordered set P of width at most w, every subset S of P satisfying |S||P| contains a chain a ₁, a ₂,..., a ₁ such that the cardinality of the interval [a _i, a _i+1] in P is the same for each i.Research supported by NSF grant DMS 8401281.Research supported by ONR grant N00014-85-K-0769.     

Countable homogeneous partially ordered sets

It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given. Research partially supported by NSF Grant MCS76-07258. Presented by B. Jónsson.     

Quasimartingales on partially ordered sets

The concept of a quasimartingale, and therefore also of a function of bounded variation, is extended to processes with a regular partially ordered index set V and with values in a Banach space. We show that quasimartingales can be described by their associated measures, defined on an inverse limit space $\text{S × Ω}$ containing $\text{V × Ω}$, furnished with the σ-algebra $\text{P}$ of the predictable sets. With the help of this measure, a Rao-Krickeberg and a Riesz decomposition is obtained, as well as a convergence theorem for quasimartingales. For a regular quasimartingale X it is proven that the spaces ($\text{S × Ω}$, $\text{P}$) and the measures associated with X are unique up to isomorphisms. In the case V = $\text{R}$₊ⁿ we prove a duality between classical (right-) quasimartingales and left-quasimartingales.     

Representation of partially ordered sets

It is well known [1] that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, (–) such that the supremum and the infimum of any finite setF ofp correspond, respectively, to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomorphic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI. Presented by R. Freese.     

Lim-inf convergence in partially ordered sets

The lim-inf convergence in a complete lattice was introduced by Scott to characterize continuous lattices. Here we introduce and study the lim-inf convergence in a partially ordered set. The main result is that for a poset P the lim-inf convergence is topological if and only if P is a continuous poset. A weaker form of lim-inf convergence in posets is also discussed.     

A construction for partially ordered sets

A construction I(S) is defined which corresponds to the intuitive notion of the set of places in which new elements can be inserted into a given poset S. It is given the minimal possible ordering. It turns out that if the base sets are chains the construction produces the corresponding interval orders. for whose dimensions there exist good estimates. In this paper we make the dual restriction that the height of the underlying set is ?1. Under this assumption we find a bound for the dimension of I(S) in general and a precise value if the set consists of two antichains all the elements of one lying above all those of the other.     

Renewal processes on partially ordered sets

Let G = (V, ≤) be a partially ordered set such that V is countable, there exists a minimum element a ∈ V, and {u ∈ V : u ≤ v} is finite for each v ∈ V. Suppose that {X_v : v ≠ a} is a collection of independent, identically distributed, nonnegative random variables. We think of X_v as the time required to perform a job associated with v, but the job at v cannot begin until all jobs at u < v are finished. The natural random variables associated with this model are the time when all jobs in a given subset of V are completed, and the set of completed jobs at a given time. These variables are studied in terms of the structure of G, and several stochastic comparison results are obtained. The common model for sequential broadcasting in trees is a special case, and in this context, we study the broadcast center of a tree. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 15–32, 2003