Common idempotents in compact left topological left semirings

摘要

A classical result of topological algebra states that any compact lefttopological semigroup has an idempotent. We refine this by showing that anycompact left topological left semiring has a common, i.e. additive andmultiplicative simultaneously, idempotent. As an application, we partiallyanswer a question related to algebraic properties of ultrafilters over naturalnumbers. Finally, we observe that similar arguments establish the existence ofcommon idempotents in much more general, non-associative universal algebras.