  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 <i>inner inverse</i>,
i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a
large class of such $x$ have inner inverses $y$ that satisfy with $x$ an
infinite family of additional monoid relations, making the monoid generated by
$x$ and $y$ what is known as an <i>inverse monoid</i> (definition recalled). We
obtain consequences of these relations, and related results.
  P. Nielsen and J. \v{S}ter, in a paper to appear, show that a much larger
class of elements $x$ of rings $R,$ including all elements of von Neumann
regular rings, have inner inverses satisfying arbitrarily large <i>finite</i>
subsets of the abovementioned set of relations. But we show by example that the
endomorphism ring of any infinite-dimensional vector space contains elements
having no inner inverse that simultaneously satisfies all those relations.
  A tangential result proved is a condition on an endomap $x$ of a set $S$ that
is necessary and sufficient for $x$ to belong to an inverse submonoid of the
monoid of all endomaps of $S.$
