Is there a version of Ore's theorem that applies to duo semigroups that are cancellative only on one side (that is, left or right cancellative)? In other words, if a duo semigroup is left or right cancellative, then is it cancellative (on both sides)? I will eventually ask a separate question on this, unless the answer turns out to be 'trivial'.

By the way, YCor's answer can be used to prove that a cancellative semigroup need not be a subdirect product of subdirectly irreducible cancellative semigroups, see mathoverflow.net/questions/480364#480364

@YCor I'm aware that group theorists tend to use the term 'bi-ordered' to mean 'totally ordered'. However, in other fields (including semigroup theory), 'ordered' is often used to mean 'partially bi-ordered'; otherwise, qualifiers like 'left' and 'right' are used, followed by terms such as 'orderable', 'totally ordered', etc. (see, for instance, Chap. 11 in the 2005 edition of Blyth's Lattices and Ordered Algebraic Structures). That's one reason why I included the definition of 'totally ordered' in my answer.

As noted by @PaceNielsen in the question linked in my previous msg, every cancellative duo embeds in a group (and hence in its universal group).

Short n' sweet.

I have the impression that the "extension lemma" in this answer holds for every duo sgrp S that embeds into its own universal group G (regardless of whether G is totally orderable and S is the associated positive cone), one key point being that, by duoness, $g^{-1}hg\in S$ for all $g,h\in S$. Here, duo means that aS = Sa for every a ∈ S. This begs another question: mathoverflow.net/q/480390/16537