(Positive) Totally Ordered Noncommutative Monoids -- How Noncommutative Can They Be?

Abstract

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition. In this note we attempt an introductory investigation of totally (also partially) ordered monoids, not assumed commutative (still writing them additively), and taking them as positive, i.e.\ every element is greater than the unit element. That, in the usual commutative cases, allows the ordering to be defined via the algebraic structure, namely, as divisibility (in our additive sense): $a\le b$ defined as $\exists\,c\,\,(b=a+c)$. The noncommutative case offers several ways to generalize that. First we try to follow the divisibility definition (on the right or on the left). Then, alternatively, we insist on the ordering being compatible with the operation both on the left and on the right, but strict inequality may not carry over -- again refer to the ordinals example. We try to see what axiom(s) such requirements impose on the monoid structure, and some facts are established. Focusing especially on the totally ordered case, one finds that necessarily the noncommutativity is somewhat limited. One may partly emulate here the commutative case, speaking about infinitely grater vs.\ Archimedean to each other elements, and in the Archimedean case even emulate Euclid's Elements' theory of `ratios' -- all that imposing some partial commutativity.