We prove that the pseudovariety of monoids of Krohn-Rhodes complexity at most n is not finitely based for all n > 0. More specifically, for each pair of positive integers n, k, we construct a monoid of complexity n+1, all of whose k-generated submonoids have complexity at most n. Mathematics Subject Classification. 20M07.

It is shown that if V is a local monoidal pseudovariety of semigroups, then K©m V, D©m V and LI©m V are local. Other operators of the form Z©m( ) are considered. In the process, results about the interplay between operators Z©m( ) and ( ) ∗ Dk are obtained.

The problem of computing the kernel of a finite monoid, popularized under the name “Rhodes type II conjecture”, led Ash to the proof of a strong property (the hyperdecidability) of the pseudovariety G of all finite groups. This paper is a survey of recent results on the study of some related properties of pseudovarieties of groups.

Non-abelian Polish groups arising as countable products of countable groups can be tame in arbitrarily complicated ways. This contrasts with some results of Solecki who revealed a very different picture in the abelian case. 0. Group trees. The class of all Polish (completely metrizable, separable) groups may be naturally divided into two classes. 0.1. Definition. A Polish group G is tame if whe...

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame ...

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame ...

Although a number of mechanical provers have been introduced and applied widely by academic researchers, these provers are rarely used in the practical development of software. For mechanical provers to be used more widely in practice, two major barriers must be overcome. First, the languages provided by the mechanical provers for expressing the required system behavior must be more natural for...

We consider tame impartial games and develop the Sprague-Grundy theory for misère playing the sum of such games that looks simpler than the classical theory suggested by Conway in 1976, which is based on the concept of genus. An impartial game is called pet if the sets of P-positions of its normal and misère versions are disjoint. We provide several equivalent characterizations and show that th...

1. Introduction. R. H. Bing showed that an upper semicontinuous decomposition of £3 into points and tame arcs did not necessarily produce a decomposition space topologically equivalent to E3 [1J. M. L. Curtis and R. L. Wilder showed [2] that the decomposition space of Bing was a homotopy manifold, thus negatively answering a question proposed by Griffiths [3J. R. Rosen recently announced [4] a ...