Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group Z0 with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to Z.

Halin proved in 1978 that there exists a normal spanning tree in every connected graph G that satisfies the following two conditions: (i) G contains no subdivision of a ‘fat’ Kא0 , one in which every edge has been replaced by uncountably many parallel edges; and (ii) G has no Kא0 subgraph. We show that the second condition is unnecessary.

We describe soluble groups in which the set of all subgroups is countable and show that locally (soluble-byfinite) groups with this property are soluble-by-finite. Further, we construct a nilpotent group with uncountably many subgroups in which the set of all abelian subgroups is countable.

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I , and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis. Let A be a finite set and let K be a f...

Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer–Specker group Zא0 with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to Z.

A conjecture of [13] states that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

We prove that a certain family of flat surfaces in genus 3 does not fulfill Veech’s Dichotomy. These flat surfaces provide uncountably many minimal but nonergodic directions. The conditions on this family are a combinatorical one and an irrationality condition. The Arnoux-Yoccoz surface fulfills this conditions.

The theory of infinite games with slightly imperfect information has been focusing on games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy Bl-ADR (as an analogue of the axiom of real determinacy ADR). We prove that the consistency strength of Bl-ADR is strictly grea...

When proving it is impossible to ‘square’ the circle by a ruler–and–compass construction we have to appeal to the theorem that π is transcendental. It is our goal to prove this theorem. Since the algebraic numbers are the roots of integer polynomials, they are countably many. Cantor’s proof in 1874 of the uncountability of the real numbers guaranteed the existence of (uncountably many) transcen...