We say that a topological monoid S is left non-archimedean (in short: l-NA) if the left action of S on itself admits a proper S-compactification \(\nu :S \hookrightarrow Y\) such that Y is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. The Stone and Pontryagin dualities play a major role in achieving useful characterizations of NA monoids. We show that many naturally defined topological monoids are NA and present universal NA monoids. Among others, we prove that the Polish monoid \(C(2^{\omega },2^{\omega })\) is a universal separable metrizable l-NA monoid and the Polish monoid \({\mathbb {N}}^{\mathbb {N}}\) is universal for separable metrizable r-NA monoids.
