Quotient sets and density recurrent sets

Quotient Sets and Density Recurrent Sets

Let S be a left amenable semigroup. Say that a subset A of S is large if there is some left invariant mean μ on S with μ(χA) > 0. A subset B of S is density recurrent if and only if, whenever A is a large subset of S, there is some x ∈ B such that x−1A ∩ A is large. We show that the set DR(S) of ultrafilters on S, every member of which is density recurrent, is a compact subsemigroup of the Ston...

Quotient Sets and Diophantine Equations

Quotient sets U/U = {u/u : u, u ∈ U} have been considered several times before in the MONTHLY [4], [5], [9]. We consider more general quotient sets U/V and we apply our results to certain simultaneous Diophantine equations with side constraints. In the following, we let R = (0,∞) denote the set of positive real numbers and N = {1, 2, . . .} the set of natural numbers. A subset S of R is said to...

Recurrent sets

A recurrent set of words is such that its minimal automaton is strongly connected. This class of sets does not seem to have been studied before. It contains the submonoids generated by prefix codes but also more general sets. We prove some properties of recurrent sets and investigate their relation with a natural generalization of prefix codes called weakly prefix codes, and also known as codes...

Recurrent and birecurrent sets

A set is called recurrent if its minimal automaton is strongly connected and birecurrent if additionally the same holds for the minimal automaton of its reversal. We study recurrent and birecurrent sets. Our main results are a new proof of an old result (the well-known theorem of Schützenberger concerning codes with bounded synchronization delay) and a series of new results concerning birecurre...

Recurrent Events and Fuzzy Sets