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.     

A reflexive digraph that is retraction trivial but not endomorphism trivial

In this note we give an example of a reflexive digraph that has no non-trivial retractions, but does have non-trivial endomorphisms.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

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 non-Bernoulli skew product which is loosely Bernoulli

LetT: Y→Y be the Bernoulli two shift with independent generatorQ={Q ₀,Q ₁} and letS: X→X be a measure preserving bijection. If (S, X) is ergodic then the skew product onX×Y defined by {fx339-1} is aK-automorphism. IfŜ is also Bernoulli we sayS is pre-Bernoulli. J. Feldman showed that ifS is pre-Bernoulli thenS must be loosely Bernoulli. We construct an example to show the converse is false, i.e. anS that is loosely Bernoulli but not pre-Bernoulli.     

A graph which is edge transitive but not arc transitive

A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.     

Epigroups in which the operation of taking pseudo-inverse is an endomorphism

The purpose of the paper is to find a characterization of the epigroups mentioned in the title in terms of forbidden divisors, which solves a problem posed by L.N. Shevrin.     

A cohesive set which is not high

We study the degrees of unsolvability of sets which are cohesive (or have weaker recursion-theoretic “smallness” properties). We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55.     

On a poset algebra which is hereditarily but not canonically well generated

LetB be a superatomic Boolean algebra.B is well generated, if it has a well founded sublatticeL such thatL generatesB. The free product of Boolean algebrasB andC is denoted byB *C. IfC is a chain thenB(C) denotes the interval algebra overC. Theorem 1: (a)Every Boolean subalgebra of B(ℵ₁) *B(ℵ₀)is well-generated. (b)B(ℵ₁) *B(ℵ₁)contains a non well-generated Boolean subalgebra. Canonical well-generatedness is defined in the introduction. Recall thatB(ℵ₁) *B(ℵ₀) is canonically well-generated, and thus well-generated. We prove the following result. Theorem 2:B(ℵ₁) *B(ℵ₀)contains a non canonically well generated Boolean subalgebra. In contrast with Theorem 1(b), we have the following result. Theorem 3:Let A ={ɑ:α<ℵ₁}⊆℘(w)be a strictly increasing sequence in the relation of almost containment. Let B be the subalgebra of ℘(w)generated by {{n}:n∈ℵ₀}∪A.Then B is superatomic, and B is not embeddable in a well-generated algebra.     

A countably categorical theory which is not G-compact

We give an example of a countably categorical theory which is not G-compact. The countable model of this theory does not have AZ-enumerations.     

A compact group which is not Valdivia compact

A compact space is Valdivia compact if it can be embedded in a Tikhonov cube in such a way that the intersection is dense in , where is the sigma-product ( the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

     

A weakly definable type which is not definable

Summary For each completion of Peano Arithmetic there is a weakly definable type which is not definable.     

An example of a function which is Denjoy integrable but not khinchin summable

The following result is proven: if ξ is an irrational number “anomalously badly“ approximable by rationals, then there are functions which are not Khinchin ξ-summable but which are Denjoy integrable. Let I be the interval 0 ≤ x ≤ 1, and let ξ be an irrational, 0 < ξ< 1. Let T _ξ denote the transformation of I into itself defined as follows: $$T_\xi x = \left\{ {\begin{array}{*{20}c} {x + \xi ,ifx + \xi \in I;} \\ {x + \xi - 1} \\ \end{array} } \right.$$ otherwise.     

A Lie grading which is not a semigroup grading

This note is devoted to the construction of a graded Lie algebra, whose grading is not given by a semigroup; thus providing a counterexample to an assertion by Patera and Zassenhaus.     

A chain compact space which is not strongly scattered

A compactT ₂ spaceX which is separable, scattered and uncountable, but still so thatX ^α−X^α+1is countable for all α∈[1, ω) is constructed. This answers one of the problems presented by M. E. Rudin in a conference as an open problem and attributed by her to Telgarsky. The author acknowledges a faculty research grant from Memphis State University when this paper was written.