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I'm trying to develop some intuition around the definition of the pseudoinverse in a regular semigroup.

Let the semigroup be $S$ with its associative operation written by juxtaposition. The pseudoinverse of an element $a \in S$ is an element $x \in S$ such that $axa = a$. A semigroup is regular if every element has at least one pseudoinverse. That's all, hopefully, standard stuff.

I'm trying to convince myself that the pseudoinverse is the "obvious" or "natural" generalization of the classical inverse.

I can get some intuition from looking at $ax$ and noting that while it may or may not be the identity, it acts like the identity on the left from the perspective of $a$. Similarly, $xa$ acts like the identity on the right of $a$.

But maybe there is something to be gleaned from maps between finite sets of the same size. Suppose that $S$ is a finite set, say $S = \{1, 2, 3\}$. Let $f$ be a function from $S$ to $S$. Now let $g : S → 2^S$ be defined as $g(x) =$ the preimage of $x$ under $f$ if that preimage is non-empty and $S$ otherwise.

Then we define a family of functions, $h_i : S → S$, where each $h_i(x)$ picks a particular element of $g(x)$. It seems to me that each $h_i$ represents something that might plausibly be called a generalized inverse. So, if an element came from a unique place, it sends it back there. If an element came from multiple places, it sends it back to one of them. If an element came from "nowhere", it sends it back to "anywhere".

It's fairly easy to check that $f(h_i(f(x))) = f(x)$ for all $i$, so that each $h_i$ is a pseudoinverse of $f$. Also, if $f$ has a classical inverse then there is only one choice of $h_i$ and that is the classical inverse.

What I'm struggling with is whether there can be any other functions that are pseudoinverses of $f$ but are not of the form $h_i$ (i.e., not generated via $g$). I.e., in the case of maps from a finite set to itself are my $g$ and $h_i$ construction and the pseudoinverse definition equivalent?

Any suggestions or pointers to the literature are welcome.

Jamie Ballingall
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2 Answers2

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I suggest looking for works from Xavier Mary who has dealt with generalised inverses. He wrote in an article in 2011: "Basically, a generalized inverse is an element that shares some (but not all) of the properties of the reciprocal inverse in a group." I found this a very nice generalisation description. His work (book-like) "Generalized inverses, semigroups and rings" is freely available.

Riivo Must
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  • Does Xavier Mary address the question of to what extent the pseudoinverse is the "obvious" or "natural" generalization? – 311411 Jan 17 '23 at 01:16
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Suppose that $h$ is a pseudo-inverse for $f$. That means $f\circ h\circ f = f$. That is, $h$ can take arbitrary values on $f(S)^c$ and $h(x)\in f^{-1}(x)$ whenever $x\in f(S)$. That is, $h(x)\in g(x)$.

So in fact, $h$ is a pseudo-inverse for $f$ iff $h(x)\in g(x)$ for all $x\in S$ as claimed.

Here $S$ can be an arbitrary set.

This should give you a pretty good intuition about pseudo-inverses, since all semigroups are subsemigroups of some semigroup of functions.

Jakobian
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