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Let $M$ be a (multiplicative) monoid with a topology $\tau$. I'd like some simple conditions for $(M,\tau)$ to be a topological monoid.

For example, a group $G$ with a topology is a topological group iff

  • the "translations" $L_a:x\in G\mapsto ax\in G$ and $R_a:x\in G\mapsto xa\in G$ are continuous,
  • the inversion $x\in G\mapsto x^{-1}\in G$ is continuous
  • and the product map $(x,y)\in G\times G\mapsto xy\in G$ in continuous at the identities $(e,e)\in G\times G$.

In the case of monoids, if we assume that the translations are open maps, then the analogous conditions to the ones above also imply that the product is continuous. But this is too strong (for example, $[0,1]$ with the operation $x\cdot y=\max(x,y)$ and the usual topology is a topological monoid, but the translations are not open).

Thanks in advance.

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There is little hope in the general case. In particular, $1$ can be the unique inversible element of a monoid (think of $\mathbb{N}$ under addition) and hence monoids can be radically different from groups. The compact case offers a few more results, but not much. Let me extract a few basic results given in the introduction of this book:

W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lecture Notes in Mathematics 1079 (1984)

Let $M$ be a compact semitopological monoid.
(1) The multiplication is jointly continuous at all points $(1,x)$ and $(x,1)$, for $x \in M$.
(2) Suppose that $M$ has a dense group of units $H$. Then for every $x \in M$ the restriction of the multiplication to the set $xM \times M$ is jointly continuous at all points $(x,s)$, for $s \in M$.

Some consequences:
(1) Every maximal subgroup of a compact semitopological semigroup is a topological group;
(2) every compact semitopological semilattice is a topological semilattice;
(3) every compact semitopological semigroup with dense subgroup contains only one minimal idempotent.

The chapter Joint continuity offers some further results that are too specialized to be reproduced here.

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