The dimension of the finite subsets κ

We answer a question of M. Pouzet by showing that the Dushnik-Miller dimension of the finite subsets of the infinite cardinal ordered by inclusion is ().This paper was written when the first author was visiting the University of Calgary, June, 1995. Research partially supported by Office of Naval Research grant NOOO14-90-1206.Research supported by NSERC grant #69-0982.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

σ-additivity in manuals and orthoalgebras

Notions of -additivity are introduced for orthoalgebras and for manuals. It is shown that the logic of a -additive manual is a -additive orthoalgebra, and that, conversely, every -additive orthoalgebra arises as such a logic. Using this theorem, it is shown that a given orthoalgebra admits at most one reasonable extension of its orthogonal sum operation to countable jointly orthogonal sets. It is also shown that every orthoalgebra can be embedded in a sigma-orthoalgebra and every orthomodular poset, in a -OMP.     

χ-Embeddings of ordered sets

We use the technique of -Embeddings to study retracts and varieties of ordered sets. We investigate the class of all ordered sets which are retract of every ordered set in which is -Embedded.     

Affine completions of distributive lattices

For any distributive lattice L we construct its extension ((L)) with the property that every isotone compatible function on L can be interpolated by a polynomial of ((L)). Further, we characterize all extensions with this property and show that our construction is in some sense the simplest possible.This research was supported by the GA SAV Grant 1230/95.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Identities of cofinal sublattices

If V is a variety of lattices and L a free lattice in V on uncountably many generators, then any cofinal sublattice of L generates all of V. On the other hand, any modular lattice without chains of order-type +1 has a cofinal distributive sublattice. More generally, if a modular lattice L has a distributive sublattice which is cofinal modulo intervals with ACC, this may be enlarged to a cofinal distributive sublattice. Examples are given showing that these existence results are sharp in several ways. Some similar results and questions on existence of cofinal sublattices with DCC are noted.This work was done while the first author was partly supported by NSF contract MCS 82-02632, and the second author by an NSF Graduate Fellowship.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On context patterns associated with concept lattices

In this paper, we consider the following reconstruction problem: Given two ordered sets (G, ) and (M, ) representing join- and meet-irreducible elements, respectively together with three relationsJ,, onG×M modelling comparability (gm) and maximal noncomparability with respect tog (gm, butgm*) and with respect tom (gm, butgm*). We determine necessary and sufficient conditions for the existence of a finite latticeL and injections :GJ(L) and :MM(L) such that the given order relations and the abstract relations coincide with the one induced by the latticeL.     

Categories of algebraic sets

The category of algebraic sets is defined in a straightforward way for any algebraic theory . It is a concrete, complete and cocomplete category dually equivalent to a full reflective subcategory of the category of -algebras. For the algebraic theory of commutative algebras over a field K, we get the algebraic sets over K.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Homothetic floating bodies

We show that a convex bodyK in ⁿ is homothetic to an ellipsoid if there is a sequence { _k }k converging to 0 so thatK is homothetic to its floating bodiesK _k.Supported by NSF grant DMS-9108003.     

Thep-adic differential-integral type operator

, p- , , , , , , . —p- - — , . , , .. , .     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

The Shifting Lemma and shifting lattice identities

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity and to any variable y occurring in we introduce a Horn sentence S(, y). When S(, y) happens to be equivalent to , we call it a shift of . When has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift.     

Free Distributive Completions of Partial Complete Lattices

The free distributive completion of a partial complete lattice is the complete lattice that it is freely generated by the partial complete lattice in the most distributive way. This can be described as being a universal solution in the sense of universal algebra. Free distributive completions generalize the constructions of tensor products and of free completely distributive complete lattices over partially ordered sets.     

A note on maximal antichains in ordered sets

We examine the question of when two consecutive levels in a product of -chains form an ordered set such that for any antichain, there is a maximal antichain disjoint from it. We characterize the pairs of consecutive levels in the product of t2 -chains that have this property. We also show that there is no upper bound on the heights of ordered sets having this property.The graph of an ordered set is the graph whose points are the elements of the ordered set, and whose edges are the ordered set's 2-element maximal antichains. We construct a class of ordered sets of all widths at least three such that the graph of each ordered set is a path, and we construct an ordered set of infinite width having a connected graph.Research supported by NSERC undergraduate student summer research fellowship, and by NSERC operating grant 69-3378Research supported by ONR contract N00014-85-K-0769