Narrow Results

Let be a class of semigroups containing all finite commutative bands, and let p(x) be a real polynomial. The -completion problem asks whether for a given partial groupoid G there exists a semigroup such that G ⊆ S, every product for (a, b) ∈ G² defined in G coincides with that for (a, b) in S, and |S| ≤ p(|G|). We prove the problem to be ℕℙ-hard in general and ℕℙ-complete if the membership problem for is in ℙ.

It is known that a poset can be embedded into a distributive lattice if, and only if, it satisfies the prime filter separation property. We describe here a class of “prime filter completions” for posets with the prime filter separation property that are completely distributive lattices generated by the poset and preserve existing finite meets and joins. The free completely distributive lattice generated by a poset can be obtained through such a prime filter completion. We also show that every completely distributive completion of a poset with the prime filter separation property is representable as a canonical extension of the poset with respect to some set of filters and ideals. The connections between the prime filter completions and canonical extensions are described and yield the following corollary: the canonical extension of any distributive lattice is the free completely distributive lattice generated by the lattice. A construction that is a variant of the prime filter completion is given that can be used to obtain the free distributive lattice generated by a poset. In addition, it is shown that every distributive lattice extension of the poset can be represented by such a construction. Finally, we show that a poset with the prime filter separation property and the free distributive lattice generated by it generates the same free completely distributive lattice.

Roger Penrose’s 2020 Nobel Prize in Physics recognizes that his identification of the concepts of “gravitational singularity” and an “incomplete, inextendible, null geodesic” is physically very important. The existence of an incomplete, inextendible, null geodesic does not say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay, we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques.

We discuss the structure of nonmonotonic reasoning. We suggest that an important aspect of nonmonotonic reasoning systems is the ordering relationship (preference structure) on the propositions in the knowledge. We suggest that this ordering relationship must be a weak order, a complete and transitive relationship. We provide a comprehensive discussion of some central issues in preference theory. In most real applications of nonmonotonic knowledge bases the ordering relationship available is often not complete. We suggest a mechanism for completing these relationships. The basic imperative used in providing this completion is to do it in such a manner as to add as few unjustified inferences as necessary. To formally accomplish this task we introduce a measure of buoyancy associated with a weak order and suggest that the preferred completion is the one with the maximal buoyancy.

Generalizing the concept of convergency to valued fields, Ostrowski in the 1930s introduced pseudo-convergent sequences. In the present paper we classify pseudo-convergent sequences in right chain domains R according to the prime ideal P associated to the breadth I of the sequence using an ideal theory developed for right cones in groups. The ring R is I-compact if every pseudo-convergent sequence in R with breadth I has a limit in R, and we construct right chain domains R which are I-compact only for right ideals I in particular subsets of the set of all right ideals of R. Krull's perfect valuation rings and then Ribenboim's notion of a valuation ring complete par étages, where is the minimal set containing the completely prime ideals in a commutative valuation ring, is a special case. For a non-discrete right invariant rank-one right chain domain R there are exactly two possibilities for the set if the value group of R is the group of real numbers under addition, and there are infinitely many possibilities for in all other cases.

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal$\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.

Let $M$ be an $R$-module over a Noetherian ring $R$ and $𝔞$ an ideal of $R$ with $c=cd(𝔞,R)$. First, over an $𝔞$-relative Cohen–Macaulay local ring $(R,𝔪)$, we provide a characterization of the $𝔞$-relative sequentially Cohen–Macaulay modules $M$ in terms of $𝔞$-relative Cohen–Macaulayness of the $R$-modules ${Ext}_R^{c-i}(M,{\text{D}}_{𝔞})$ for all $i\ge 0$, where ${\text{D}}_{𝔞}={Hom}_R(H_{𝔞}^c(R),E(R/𝔪))$. Next, we prove that $M$ is finite $𝔞$-relative Cohen–Macaulay if and only if $H_i(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^{cd(𝔞,M)}(M)))=0$ for all $i\ne cd(𝔞,M)$ and $H_{cd(𝔞,M)}(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^{cd(𝔞,M)}(M)))\cong {\widehat{M}}^{𝔞}$. Finally, we provide another characterization of $𝔞$-relative sequentially Cohen–Macaulay modules $M$ in terms of vanishing of the local homology modules $H_j(L{\mathrm{\Lambda }}_{𝔞}(H_{𝔞}^i(M)))=0$ for all $0\le i\le cd(𝔞,M)$ and for all $j\ne i$.

The construction of wavelet functions from known scaling functions is called the 'completion problem'. Completion algorithms exist for univariate wavelets, including multiwavelets. For multivariate wavelets, however, completion is not always possible. We present a new algorithm (a generalization of a method of Lai) which works in many cases.

This paper is concerned with compactifications of high-dimensional manifolds. Siebenmann’s iconic 1965 dissertation [L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton Univ. (1965), MR 2615648] provided necessary and sufficient conditions for an open manifold $M^m$ ($m\ge 6$) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where $M^m$ is noncompact with compact boundary; however, when $\partial M^m$ is noncompact, the situation is more complicated. The goal becomes a “completion” of $M^m$, i.e. a compact manifold ${\widehat{M}}^m$ containing a compactum $A\subseteq \partial M^m$ such that ${\widehat{M}}^m\setminus A\approx M^m$. Siebenmann did some initial work on this topic, and O’Brien [G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl.16 (1983) 303–324, MR 722123] extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here, we provide such a characterization. Our second main theorem involves ${𝒵}$-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann [T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math.137 (1976) 171–208, MR 0425973] guarantee ${𝒵}$-compactifiability for a manifold $M^m$. We cannot answer that question, but we do show that those conditions are satisfied if and only if $M^m\times [0,1]$ is ${𝒵}$-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem — an application that partly explains our current interest in that topic, and also illustrates the utility of the ${\pi }_1$-condition found in that theorem.

In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.

Many methods deal with missing values, mainly focused on their completion. However, they complete indifferently all the missing values regardless of their origin, i.e., they assume that all the missing values occur randomly in a dataset. In this paper, we show that many missing values do not stem from randomness, We use the relationships within the data and define four types of missing values. The characterization is made for each missing value. We claim that such a local characterization enables us perceptive techniques to deal with missing values according to their origins. Next, we show how this typology is suitable for completing the missing values: it considers the non-randomness appearance of the missing values and suggests the values of completion.