Retract extensions of lattices

As a continuation of the paper on the ideal extensions of lattices considered in Russian Mathematics (Iz. VUZ) 53 (2), 41–57 (2009), we present here a detailed exposition of the retract extensions of lattices.     

Minimizing bumps in linear extensions of ordered sets

A linear extension x ₁ x ₂ x ₃ ... of a partially ordered set (X, <) has a bump whenever x _i <x _i +1. We examine the problem of determining linear extensions with as few bumps as possible. Heuristic algorithms for approximate bump minimization are considered.     

Retract rational fields and cyclic Galois extensions

In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field. The author is grateful for NSF support under grant #MCS79-04473.     

On some new types of greedy chains and greedy linear extensions of partially ordered sets

Two new types of greedy chains, strongly and semi-strongly greedy, in posets are defined and their role in solving the jump number problem is discussed in this paper. If a poset P contains a strongly greedy chain C then C may be taken as the first chain in an optimal linear extension of P. If a poset P has no strongly greedy chains then it contains an optimal linear extension which starts with a semi-strongly greedy chain. Hence, every poset has an optimal linear extension which consists of strongly and semi-strongly greedy chains. Algorithmic issues of finding such linear extensions are discussed elsewhere (Syslo, 1987, 1988), where we provide a very efficient method for solving the jump number problem which is polynomial in the class of posets whose arc representations contain a bounded number of dummy arcs. In another work, the author has recently demonstrated that this method restricted to interval orders gives rise to 3/2-approximation algorithm for such posets.     

Powers of ordered sets

Recently there has been significant progress in the study of powers of ordered sets. Much of this work has concerned cancellation laws for powers and uses these two steps. First, logarithmic operators are introduced to transform cancellation problems for powers into questions involving direct product decompositions. Second, refinement theorems for direct product decompositions are brought to bear. Here we present two results with the aim of highlighting these steps.Supported by NSF grant MCS 83-02054     

Ladders in ordered sets

An ordered set P is said to have the 2-cutset property if for every element x of P there is a subset S of P whose elements are noncomparable to x, such that |S|2 and such that every maximal chain in P meets {x}S. It is shown that if P has the 2-cutset property and has width n then P contains a ladder of length [1/2(n–3)].     

Totally ordered uncertain sets

It is known that some uncertain sets have membership functions, and some do not. How do we judge whether an uncertain set has a membership function? In order to answer this question, this paper presents a concept of totally ordered uncertain set, and shows that totally ordered uncertain sets always have membership functions if they are defined on a continuous uncertainty space. In addition, some criteria for judging the existence of membership functions for uncertain sets are provided. Several inspiring examples and counterexamples are also documented in this paper.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.     

Finite projective ordered sets

We show that quasiprojectivity and projectivity are equivalent properties for finite ordered sets of more than two elements.     

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.     

Holes in ordered sets

We pursue the technique of “holes” to study the retracts of an ordered set. This is applied to establish a close connection between the class of absolute retracts and the class of dismantlable 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.     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

Nonstandard analysis of ordered sets

We introduce a nonstandard approach to the study of ordered setsX based on a classification of the elements of the ordered set *X into three types, upward, downward, and lateral, which may be thought of dynamically as arising from the possibilities of upward, downward, and lateral motion withinX. Initial applications include the characterization thatX has no infinite diverse subset iff *X has no lateral elements, a result subsequently exploited in work on the interval topology and order-compatibility, where we give a nonstandard proof of Naito's result that ifX has no infinite diverse subset, it has a unique order-compatible topology. We also describe how the completion of a nonempty linearly ordered setX may be obtained as a quotient of *X.