Common idempotents in compact left topological left semirings

TL;DR

This paper proves that compact left topological left semirings always contain a common idempotent element for both addition and multiplication, extending classical results and addressing open questions in algebraic structures.

Contribution

It introduces the existence of common idempotents in compact left topological left semirings, generalizing previous results and applying to ultrafilters and non-associative algebras.

Findings

Existence of common idempotents in compact left topological left semirings

Partial answer to algebraic properties of ultrafilters over natural numbers

Extension of arguments to non-associative universal algebras

Abstract

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