On a Generalization of a Result by Valk and Jantzen

International Journal of Quantum Chemistry, 1976

Liiwdin has formulated a useful criterion for testing the completeness of an expansion basis set. From this criterion one may obtain an incompleteness coefficient for a truncated (finite) basis set. We have investigated the significance of this incompleteness coefficient for some of the basis sets we have used in our recent calculations of bounds to quantum mechanical properties. The similarity of the convergence properties between our bounds and the incompleteness coefficient suggests that Lowdin's criterion is likely to be useful in practice.

Journal of the Australian Mathematical Society, 1987

Several “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes w...

Well-Quasi Orders in Computation, Logic, Language and Reasoning, 2020

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

Order, 1997

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. In particular, if K(P) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.

Archive for Mathematical Logic, 2017

As the class PCSL of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a ℵ 0 -categorical model companion PCSL * . As PCSL is inductive the models of PCSL * are exactly the existentially closed models of PCSL. We will construct the unique existentially closed countable model of PCSL as a direct limit of algebraically closed pseudocomplemented semilattices.

Mathematica Slovaca, 2007

In this paper we define partially ordered quasi-uniform spaces (X, $$\mathfrak{U}$$ , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity $$\mathfrak{U}$$ on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology $$\tau _{\mathfrak{U}*} $$ of a PO-quasi-uniform space (X, $$\mathfrak{U}$$ , ≤), the bicompletion $$(\tilde X,\tilde {\mathfrak{U}})$$ of (X, $$\mathfrak{U}$$ ) is also a PO-quasi-uniform space ( $$(\tilde X,\tilde {\mathfrak{U}})$$ , ⪯) with a partial order ⪯ on $$\tilde X$$ that extends ≤ in a natural way.

Annals of Pure and Applied Logic, 2004

Wilkie (Selecta Math. (N.S.) 5 (1999) 397) proved a "theorem of the complement" which implies that in order to establish the o-minimality of an expansion of R with C ∞ functions it su ces to obtain uniform (in the parameters) bounds on the number of connected components of quantiÿer free deÿnable sets. He deduced that any expansion of R with a family of Pfa an functions is o-minimal. We prove an e ective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered ÿeld R with ÿnitely many C ∞ functions, if there are uniform and computable upper bounds on the number of connected components of quantiÿer free deÿnable sets, then there are uniform and computable bounds for all deÿnable sets. In such a case the theory of the structure is e ectively o-minimal: there is a recursively axiomatized subtheory such that each of its models is o-minimal. This implies the e ective o-minimality of any expansion of R with Pfa an functions. We apply our results to the open problem of the decidability of the theory of the real ÿeld with the exponential function. We show that the decidability is implied by a positive answer to the following problem (raised by van den Dries

International Journal of Algebra, 2012

This investigation is in the Bishop's constructive mathematics (in the sense of Bishop, Bridges, Richman, Ruitenburg, Dirk van Dalen and AS Troelstra). The main subject of this consideration is a construction of quasi-antiorder relation on semigroup with ...

Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as "bubbles" (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles. As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.

Topology and its Applications, 2001

We prove the Countable Extension Basis (CEB) Theorem for noncompact metric spaces. As an application of the CEB Theorem we give alternative proofs of some classical results in the area. In particular, we construct a universal complete metric space X K in the class of spaces with e-dim X K together with a K-soft map onto the Hilbert cube. As a corollary we obtain Olszewski's completion theorem. Also we apply the CEB Theorem to give an alternative proof of the Splitting Theorem in extension theory for noncompact spaces. 

The electronic journal of combinatorics

We consider conditions which force a well-quasi-ordered poset (wqo) to be better-quasi-ordered (bqo). In particular we obtain that if a poset $P$ is wqo and the set $S_{\omega}(P)$ of strictly increasing sequences of elements of $P$ is bqo under domination, then $P$ is bqo. As a consequence, we get the same conclusion if $S_{\omega} (P)$ is replaced by $\mathcal J^1(P)$, the collection of non-principal ideals of $P$, or by $AM(P)$, the collection of maximal antichains of $P$ ordered by domination. It then follows that an interval order which is wqo is in fact bqo.

Sibirskie Elektronnye Matematicheskie Izvestiya, 2021

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of k-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to k-partitions.

Mathematica Pannonica

The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.

Annals of Pure and Applied Logic, 2005

In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, a so-called pseudo-order. The prime example is the order of the constructive real numbers. We examine two kinds of constructive completions of pseudo-orders: order completions of pseudo-orders and Cauchy completions of (non-archimedean) ordered groups and fields. It is shown how these can be predicatively defined in type theory, also when the underlying set is non-discrete. Provable choice principles, in particular a generalisation of dependent choice, are used for showing set-representability of cuts.

1999

A natural strengthening of the completeness hypothesis, called *-completeness, allows us to extend to ωµ-metric spaces some properties of complete metric spaces which fail to hold if only the completeness condition is assumed. In this way we achieve some results on hyperspace theory, dealing with supercompleteness and with selections of multivalued functions. The notion of *-completeness is investigated and it is proved that quite important classes of ωµ-metric spaces have this property.