测度拓扑和连续偏序集的刻画

研究偏序集上的测度拓扑以及与其它内蕴拓扑间的关系,利用测度拓扑刻画了偏序集的连续性.构造了反例说明存在完全分配格,其上的测度拓扑不是连续格从而不是局部紧拓扑.     

Strongly continuous posets and the local Scott topology

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SC-posets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LC-continuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T₀-space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order.     

Ultraproducts and Chevalley groups

Given a simple non-trivial finite-dimensional Lie algebra L, fields and Chevalley groups , we first prove that is isomorphic to . Then we consider the case of Chevalley groups of twisted type . We obtain a result analogous to the previous one. Given perfect fields having the property that any element is either a square or the opposite of a square and Chevalley groups , then is isomorphic to . We apply our results to prove the decidability of the set of sentences true in almost all finite groups of the form L(K) where K is a finite field and L a fixed untwisted Chevalley type. Received: 19 November 1993 / Revised version: 15 November 1995     

Ultraproducts preserve finite subdirect reducibility

An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties.     

Ultraproducts and ultra-limits of near-rings

The construction of ultraproducts, ultrapowers and ultralimits have proven useful, both for logic and for special (algebraic) structures. It seems that these constructions might be especially useful in studying near-rings; they might provide solutions to long-standing problems. Among other results, we show that ultraproducts of primitive near rings are again primitive.Part of this work was done while the second author was a Visiting Professor at the University of Southwestern Louisiana (Lafayette, La). This author expresses his gratitude for the received hospitality and for most valuable discussions at this Department. Special thanks are due to Professors D. Blumberg, H. Heatherly and A. Iskander.     

Ultraproducts and Higher Order Formulas

Which ultraproducts preserve the validity of formulas of higher order logics? To answer this question, we will introduce natural topologies (which we call ultratopologies) on ultraproducts. We will show, that ultraproducts preserving certain higher order formulas can be characterized in terms of these topologies. As an application of the above results, we provide a constructive, purely model theoretic characterization for classes definable by second order existential formulas.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Schedule-induced posets

Schedules are a basic tool in the treatment of project networks and sequencing problems. For many questions, however, it is not the schedule itself but a partial order (poset) naturally induced on the set of activities that is really relied on. In fact, this correspondence of the usually uncountably many schedules to a finite system of posets was a key to many recent results in this field and constitutes the discrete character of the described problems, all the more so as additional properties of the respective schedules can very often be reflected in related properties of these induced posets.The present paper gives insights into the class of those posets that can be induced by schedules. An immediate observation is that schedule-induced posets are just the interval orders. More involved characterizations are concerned with the possible ‘local optimality’ of such structures. One such property is the existence of activity durations that will result in a project duration of the considered poset that is shorter (simultaneously) than the duration of all its extensions. A similar, related characterization for general regular cost functions (in particular tardiness cost) is also included, as are insights into ‘best’ activity durations with the mentioned properties.The paper closes with some remarks and numerical data on the considered class of posets as a whole.     

Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

N-free posets as generalizations of series-parallel posets

N-Free posets have recently taken some importance and motivated many studies. This class of posets introduced by Grillet [8] and Heuchenne [11] are very related to another important class of posets, namely the series-parallel posets, introduced by Lawler [12] and studied by Valdes et al. [21]. This paper shows how N-free posets can be considered as generalizations of series-parallel posets, by giving a recursive construction of N-free posets. Furthermore we propose a linear time algorithm to recognize and decompose any N-free poset. This yields some very naturel problems, namely: which are the properties(such as linear time algorithm for some invariant) of series-parallel posets that are kept for N-free posets?     

Incidence posets of trees in posets of large dimension

In this paper, we investigate substructures of partially ordered sets which must be present whenever the dimension is large. We show that for eachn1, ifT is a tree onn vertices and ifP is any poset having dimension at least 4n ⁶, then eitherP or its dual contains the incidence poset ofT as a suborder.     

Realizable posets

Certain p-local orders in n-dimensional division algebras over the rational numbers occur as endomorphism rings of torsion-free abelian groups of rank n if and only if an associated finite poset P has a strict faithful representation of dimension less than |P| over the field with p elements. In this note we obtain a simple characterization of those finite posets which do not admit such a representation.Research supported in part by NSF grant DMS-8802833.     

Quasicontinuous posets

Communicated by Michael Mislove     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.