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Recently, a general version of the Hoffmann-Jorgensen inequality was shownjointly with Rajaratnam [Ann. Probab. 2017], which (a) improved the result evenfor real-valued variables, but also (b) simultaneously unified and extendedseveral versions in the Banach space literature, including that by Hitczenkoand Montgomery-Smith [Ann. Probab. 2001], as well as special cases and variantsof results by Johnson-Schechtman [Ann. Probab. 1989] and Klass-Nowicki [Ann.Probab. 2000], in addition to the original versions by Kahane andHoffmann-Jorgensen. Moreover, our result with Rajaratnam was in a primitiveframework: over all semigroups with a bi-invariant metric; this includes Banachspaces as well as compact and abelian Lie groups.  In this note we show the result even more generally: over every semigroup$\mathscr{G}$ with a strongly left- (or right-)invariant metric. We also provesome applications of this inequality over such $\mathscr{G}$, extending Banachspace-valued versions by Hitczenko and Montgomery-Smith [Ann. Probab. 2001] andby Hoffmann-Jorgensen [Studia Math. 1974]. Furthermore, we show several otherstochastic inequalities - by Ottaviani-Skorohod, Mogul'skii, and Levy-Ottaviani- as well as Levy's equivalence, again over $\mathscr{G}$ as above. Thissetting of generality for $\mathscr{G}$ subsumes not only semigroups withbi-invariant metric (thus extending the previously shown results), but it alsomeans that these results now hold over all Lie groups (equipped with aleft-invariant Riemannian metric).  We also explain why this primitive setting of strongly left/right-invariantmetric semigroups $\mathscr{G}$ is equivalent to that of left/right-invariantmetric monoids $\mathscr{G}_\circ$: each such $\mathscr{G}$ embeds in some$\mathscr{G}_\circ$.

Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups