# Conditions for Monoid to be a topological monoid

Let $M$ be a (multiplicative) monoid with a topology $\tau$. I'd like some simple equivalent 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 operation $(x,y)\in G\times G\mapsto xy\in G$ in continuous in the identities $(e,e)\in G\times G$.

In the case of monoids, if we assume that the translations are open mappings then the same arguments as in the case of groups can be used. But I think that this condition is way too strong. I don't want to use the inverse element also.

Thanks.

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