  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.
