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Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is linearly orderable if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < cb$ for all $a,b,c \in A$ with $a < b$ (note strict inequalities).

Some examples of linearly orderable semigroups are: the real numbers with the usual addition; the positive integers divisible only for the members of a given set $S$ of (natural) primes with the usual multiplication; every abelian torsion-free cancellative semigroup (see Proposition 2 below); the polynomials in finitely many variables with nonnegative real coefficients with the usual Cauchy multiplication; the upper [lower] triangular matrices with positive real entries and the usual row-by-column multiplication; the free monoid on an alphabet $X$; subsemigroups and direct products of the previous ones.

Now, denote by $\mathfrak A^{(1)}$ the (canonical) unitization of $\mathfrak A$. As a byproduct of something that I've just finished to write, I happened to prove the following:

Proposition 1. $\mathfrak A$ is linearly orderable (if and) only if the same holds true with $\mathfrak A^{(1)}$.

Proposition 2. Every abelian torsion-free (*) cancellative semigroup $\mathfrak A$ is linearly orderable.

Proposition 2 has a kind of (trivial) converse: Every linearly orderable semigroup is torsion-free and cancellative (indeed, something stronger can be proved; i.e., none of the elements of the semigroup has finite order unless the semigroup is unital and such an element is the identity).

I am reasonably sure that both results are nothing new, but I wasn't able to find any reference. In particular, I checked Clifford's 1958 survey, but this seems focused more on totally ordered semigroups (there referred to simply as ordered semigroups) than on linearly ordered semigroups (there called strictly ordered semigroups). On another hand, I am aware of a 1913 result by F.H. Levi (Arithmetische Gesetze im Gebiete diskreter Gruppen, Rend. Circ. Mat. Palermo, Vol. 35 (1913), pp. 225-236), where it is proved that every torsion-free abelian group is linearly orderable (as a group). On another hand, I have no clue about Proposition 1. Then, here are my questions:

Question 1. Do you know of any paper, book, comic strip (I'm damned serious) with a published proof of Propositions 1 and/or 2?

Question 2. Any hint on how to retrieve Levi's original paper? It seems impossible to find it, and there is no copy of it in my local library.

Thank you in advance.


(*) To avoid misunderstandings due to terminology, I say that a semigroup $\mathfrak A$ is torsion-free if an element $a$ has finite order, that is, $a^m = a^n$ for some $m,n \in \mathbb N^+$ with $m \ne n$, if and only if $a$ is idempotent.

Extra contents. For what it is worth, my proof of Proposition 2 does not really add any significant new idea; it is based on Levi's result and use nothing but well-known basic facts: (i) $\mathfrak A$ embeds in its unitization $\mathfrak A^{(1)}$; (ii) $\mathfrak A$ is abelian/cancellative/torsion-free iff the same holds true with $\mathfrak A^{(1)}$; (iii) the inverse image of a linearly orderable semigroup under a semigroup embedding is linearly orderable; (iv) every subsemigroup of a linearly orderable semigroup is itself linearly orderable; (v) as a consequence of (i)-(iv), we can assume wlog that $\mathfrak A$ is an abelian torsion-free cancellative monoid and construct its Grothendieck group, say $\mathfrak A_\mathcal{G}$; (vi) $\mathfrak A_\mathcal{G}$ is torsion-free (and obviously abelian); (vii) $\mathfrak A$ embeds in $\mathfrak A_\mathcal{G}$, by cancellativity; (viii) we can use (iii), (iv), (vii) and Levi's result to conclude. Nonetheless, I think that it may deserve a little place in the paper (e.g., as a reference for future work). But I would feel better if I could have a pointer to a previously published proof. It goes the same with Proposition 1.

Motivation (if it matters; if not, ignore it all). Freiman, Herzog and coauthors have recently proved some results on sum-sets/produc-sets in linearly ordered groups (which they refer to simply as ordered groups), accordingly extending some parts of Freiman's previous work on small doubling on integers; see G. Freiman, M. Herzog, P. Longobardi, and M. Maj, Small doubling in ordered groups, J. Austral. Math. Soc., to appear. Even more recently, I myself extended some of their results from the setting of linearly ordered groups to linearly ordered semigroups. And this is where my questions come from.

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You are right that it is nothing new. I had the same proof right away. I am not sure where it is first written. If I have a moment I can try to dig up a reference. –  Benjamin Steinberg Aug 29 '12 at 19:57
Look also at mathoverflow.net/questions/92818/… –  Benjamin Steinberg Aug 29 '12 at 20:00
Btw strict and not strict orderability are the same for cancelative semigroups so Clifford should have it. –  Benjamin Steinberg Aug 29 '12 at 20:05
I agree, my comment should be referred to both propositions, sorry for the lack of clarity. As for the second one, Clifford doesn't, however, seem to address torsion-freeness at all, unless I'm missing something. –  Salvo Tringali Aug 29 '12 at 22:28
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1 Answer

Corollary 3.4 of Gilmer's book on commutative semigroup rings states a commutative semigroup is totally orderable (in your sense) iff torsion-free and cancellative.

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Great! Thank you, @Benjamin. This settles the matter with Proposition 2. Any clue about the other? –  Salvo Tringali Aug 29 '12 at 22:35
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