A not so simple local multiplier algebra

We construct an AF-algebra A such that its local multiplier algebra Mloc(A) does not agree with Mloc(Mloc(A)), thus answering a question raised by G.K. Pedersen in 1978.     

Boundedly isometric but not isometric spaces

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 3, pp. 118–121, May–June, 1993.     

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.      

Rationally but not integrally isometric bilinear forms

Let K be a non-dyadic local field and A its ring of integers. First we notice that the classification of unimodular A-bilinear forms and of systems of two unimodular symmetric A-bilinear forms are both reduced to the classification of hermitian forms over the category of double isomorphisms over projective A-modules ; the duality is nevertheless different for each case. Then we use the methods of reduction and transfer to construct some examples of non isometric unimodular forms (or system of unimodular symmetric forms) having the same asymmetry and becoming isometric over K.     

Weakly representable but not representable relation algebras

We prove the existence of non-representable relation algebras the union and complementation free reducts of which can be represented, i.e. which are weakly representable. This answers Problem 3 in Jónsson [4], and has consequences concerning the complexity of the equational theory of representable relation algebras.Presented by B. Jonsson.Research supported by Hungarian National Foundation for Scientific Research grants No. 1911 and No. T7255.     

Smooth,but not complex-analytic pluripolar sets

It is shown that the exact --sets of plurisubharmonic functions are not necessarily complex-analytic even if they are closed C -smooth real submanifolds.     

Isomorphic but not base-isomorphic base-minimal cylindric set algebras

We show the existence of finitely-generated, isomorphic but not base-isomorphic base-minimal cylindric set algebras. These algebras are regular and locally finite. This solves a problem mentioned in both books [HMTAN 81] and [HMT II].Presented by Jan Mycielski.Research supported by Hungarian National Foundation for Scientific Research grant no. 1810.     

Connected but not path-connected subspaces of infinite graphs

Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure.     

Flexible Majority Rules in democracyville: A guided tour

“Flexible Majority Rules” are rules for which the majority threshold depends on the proposal on the table. We present the core ideas of Flexible Majority Rules with a new model and survey the current state of affairs of such rules. We show that the proposal-dependency of Flexible Majority Rules can take many forms and it allows many applications. Finally, we outline new research areas and applications for Flexible Majority Rules.     

Full but not strong dualities at the finite level: Extending the realm

The realm of natural dualities that are known to be full but not strong at the finite level is a very small one, consisting of a single example. This example, based on the three-element bounded distributive lattice, was presented by Davey, Haviar and Willard [8]. In this paper, we extend this realm to the class of all natural dualities based on a finite non-boolean bounded distributive lattice. Received June 15, 2005; accepted in final form November 26, 2005.     

A Banach space that is MLUR but not HR

Mathematische Annalen -     

Continua that are shape-equivalent but not UV1-equivalent

It is shown that each continuum X with pro-π₁(X) not pro-finite admits uncountably many continua that are shape-equivalent to X but pairwise UV¹-inequivalent.     

A dyadic endomorphism which is Bernoulli but not standard

Any measure preserving endomorphism generates both a decreasing sequence of σ-algebras and an invertible extension. In this paper we exhibit a dyadic measure preserving endomorphism (X,T,μ) such that the decreasing sequence of σ-algebras that it generates is not isomorphic to the standard decreasing sequence of σ-algebras. However the invertible extension is isomorphic to the Bernoulli two shift.     

A quadratic field which is Euclidean but not norm-Euclidean

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of .     

A markov random field which isK but not Bernoulli

We use a variant of a discrete time exclusion process, studied by Yaguchi [7], to construct a Markov random field which isK but not Bernoulli. Instead of having all the particles in the exclusion process indistinguishable, this system has two different types of particles.