  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 $\nu
\colon S \hookrightarrow Y$ 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 major role in achieving useful
characterizations of NA monoids. We discuss universal NA monoids and show that
many naturally defined topological monoids are NA. 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.
