We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields such as (i) G is definably an almost direct product of a semisimple group and a commutative group, (ii) (G, ·) is elementarily equivalent to (G/G, ·). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as deducing the full compact domin...

In 1914 Lebesgue defined a ‘universal covering’ to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pál, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen’...

K e y w o r d s S e t c o v e r i n g problems, Fuzzy sets, Fuzzy set-covering problem, Algebraic sum operator. 1. I N T R O D U C T I O N A classical set-covering problem considers the subsets I = {1, 2 , . . . , m} and J = {1, 2 , . . . , n} of integers. A collection p of subset Pj of I is a cover of I, if the union of the members of go is I, i.e., let Pj C I, go = {Pj : j C J0}, J0 C J, go i...

We present a notion of forcing that can be used, in conjunction with other results, to show that there is a Martin-Löf random set X such that X 6≥T ∅′ and X computes every K-trivial set.