Non-archimedean topological monoids

Abstract 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 $$\nu :S \hookrightarrow Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>:</mml:mo> <mml:mi>S</mml:mi> <mml:mo>↪</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> 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 $$C(2^{\omega },2^{\omega })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>ω</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>ω</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a universal separable metrizable l-NA monoid and the Polish monoid $${\mathbb {N}}^{\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> is universal for separable metrizable r-NA monoids.