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For $V$ a vector space over a field, or more generally, over a division ring,it is well-known that every $x\in\mathrm{End}(V)$ has an inner inverse,i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that alarge class of such $x$ have inner inverses $y$ that satisfy with $x$ aninfinite family of additional monoid relations, making the monoid generated by$x$ and $y$ what is known as an inverse monoid (definition recalled). Weobtain consequences of these relations, and related results. P. Nielsen and J. \v{S}ter, in a paper to appear, show that a much largerclass of elements $x$ of rings $R,$ including all elements of von Neumannregular rings, have inner inverses satisfying arbitrarily large finitesubsets of the abovementioned set of relations. But we show by example that theendomorphism ring of any infinite-dimensional vector space contains elementshaving no inner inverse that simultaneously satisfies all those relations. A tangential result proved is a condition on an endomap $x$ of a set $S$ thatis necessary and sufficient for $x$ to belong to an inverse submonoid of themonoid of all endomaps of $S.$

Strong inner inverses in endomorphism rings of vector spaces