For V a vector space over a field, or more generally, over a division ring,
it is well-known that every x∈End(V) has an inner inverse,
i.e., an element y∈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 inverse monoid (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 finite
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.Comment: 18pp. The main change from the preceding version is the discussion of
three questions posed by the referee, two on p.10, starting on line 6, and
one starting at the top of p.16. There are also many small revisions of
wording et
