Sums of idempotents in \(\beta{}\mathbb{N}\)
Let + also indicate the continuous extension to \(\beta\mathbb{N}\) of addition on \(\mathbb{N}\). The structure theorem for compact semigroups tells us that the sum \(p+q\) of two idempotents in the same minimal left (or right) ideal of \(\beta\mathbb{N}\) is also an idempotent. Here the authors address the question: is the sum \(p+q\) of minimal idempotents an idempotent only in this situation? In their main result they find a family \(\{p_ \gamma\mid \gamma < 2^ c\}\) of minimal idempotents such that \(p_ \gamma+p_ \mu\) and \(p_ \mu+p_ \gamma\) are not idempotent whenever \(\gamma < \mu < 2^ c\).
scientific article; zbMATH DE number 34206