Strong inner inverses in endomorphism rings of vector spaces

TL;DR

This paper investigates conditions under which endomorphisms of vector spaces possess inner inverses that generate inverse monoids, revealing that some infinite-dimensional cases lack such universal inverses despite known existence of inner inverses.

Contribution

It demonstrates that certain endomorphisms in infinite-dimensional vector spaces have inner inverses satisfying extensive monoid relations, expanding understanding of inverse monoids in endomorphism rings.

Findings

Some endomorphisms lack inner inverses satisfying all relations.

Finite-dimensional cases always have such inverse monoids.

Infinite-dimensional cases can lack universal inverse monoids.

Abstract

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 $S.$