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The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of monoids with this property:

Incidentally, the three classes of monoids from this list have, of course, non-empty intersection, but none of them is contained in the union of the others.

Edit. In a previous version of this post, I was writing that the property in the title holds if $H^\times = \{1_H\}$ (i.e., for reduced monoids). That's not true, as can be seen by looking at $\langle x, y \mid xy = 1 \rangle$.

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    $\begingroup$ These are monoids that do not contain the bicycling monoid as a submonoid. $\endgroup$
    – Benjamin Steinberg
    Commented Feb 11, 2017 at 21:08
  • $\begingroup$ I hope you'll forgive my ignorance, what is the bicycling monoid? A Google search didn't help me much. Edit. OK, I guess you mean the bicyclic monoid from p. 32 in the 2003 reprint of Howie's Fundamentals of Semigroup Theory. $\endgroup$
    – Salvo Tringali
    Commented Feb 11, 2017 at 21:27
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    $\begingroup$ Sorry my phone changed bicyclic to bicycling $\endgroup$
    – Benjamin Steinberg
    Commented Feb 11, 2017 at 22:21

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The correct term is Dedekind finite. A monoid is Dedekind finite of $xy=1$ implies $yx=1$. This is clearly equivalent to your condition.

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    $\begingroup$ Cool. In particular, if $xy \in H^\times$ for some $x, y \in H$, then there exists $z \in H$ for which $xyz = zxy =1_H$. So, if $H$ is Dedekind-finite, then $yzx = 1_H$, hence $x$ and $y$ are invertible. $\endgroup$
    – Salvo Tringali
    Commented Feb 12, 2017 at 9:47

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