This is not, perhaps, a very large area, nor a complete "ending", but it was an interesting development in early semigroup theory that I think bears writing down.

Some background, first. A semigroup $S$ is a set with an associative binary operation $\cdot : S \times S \to S$. A semigroup is *left cancellative* if for all $a, b, c \in S$, we have $ab = ac$ implies $b = c$, and *right cancellative* if $ba = ca$ implies $b = c$. A semigroup is *cancellative* if it is left and right cancellative.

All groups are cancellative semigroups, but there are cancellative semigroups which are not groups (free semigroups, for example). Hence being cancellative is a necessary condition for a semigroup to embed in some group. A natural question is the following: **does every cancellative semigroup embed in a group?**

Anton Sushkevich initiated the study of cancellative semigroups in 1928. He was very interested in the problem of embedding cancellative semigroups in groups, and predicted that this very problem would become a central part of semigroup theory and produce a vast amount of new results in the area. This problem led to several publications by him and several others over the next few years, developing the theory of embedding cancellative semigroups in groups.

In [A. Sushkevich, "Про поширення півгрупи до цілої группы", Zapiski Khark. Mat. 4:12 (1935)], Sushkevich claimed a full affirmative answer -- being cancellative, he claimed, is sufficient for a semigroup to embed in a group!

But alas, in 1937, Malcev proved by way of example that there exists a cancellative semigroup which does not embed inside a group! In fact, he even provided a countable list of necessary and sufficient conditions for a cancellative semigroup to embed in a group, and showed that no finite sublist of this list is sufficient [A. Malcev, "On the Immersion of an Algebraic Ring into a Field," Math Annalen, Bd. 113, 5 Heft (1937)].

And yet, Sushkevich published a book in 1937, "The Theory of Generalised Groups" (of which very few physical copies still remain due to most copies being stored in Kharkov, Ukraine, during WW2, a city which was ruined in numerous battles during the war), which claimed to fix the errors in his original proof. The proof is tricky to read -- and while Sushkevich does acknowledge Malcev's example, he more or less only says "and so a sufficient condition for a semigroup to embed in a group is that it is cancellative *and that it is not Malcev's example*.

Malcev turned out to be right, and Sushkevich's proof was wrong (the operation for the "group" he claims to embed the semigroup in is not associative!). Indeed, Sushkevich spent the next few years attempting to remove any trace of his original publication, to the point where I would be surprised if any copy of the paper could be found (**edit:** a few years later, I was surprised, see below). So a single counterexample was enough to -- if not "close" -- at least "deflate" the hype around cancellative semigroups.

Of course, as with any result, it only really served to spur further refinements and new lists of sufficient and necessary conditions, but its predicted central importance to semigroup theory fell short, and semigroup theorists mostly moved on to new pastures.

**Update:** In early 2023, I translated all of Suskevich's Ukrainian articles into English, and posted the translation on arXiv:2302.02996, including his "lost" 1935 article (C. Hollings sent me a copy) with the above result. More details and references can be found there.