Right-orderability versus left-orderability for monoids

International audience; We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and counterexamples, also highlighting the particular role played by \emph{positive} orderability.Moreover, we construct a non-left-orderable, positively right-orderable submonoid of the free product of the cyclic group of order 7 with the free group on four generators.Any group extension of that monoid has elements of order 7.