Smallest non-commutative totally ordered semigroup

We considered commutative aperiodic semigroups such as semilattices upper bound functions on partial orders. We even considered how these upper bound functions can themselves be partially ordered by "how upper" their bounds are. Now we will move to the non-commutative case. Towards that end, we should consider the simplest case of a non-commutative upper bound function:

[[1 1 1]
 [1 1 1]
 [1 2 3]]

The first thing we notice is that this semigroup is actually totally ordered [3,2,1]. The second thing that we notice about this semigroup is that the middle element is index two. Therefore, the corresponding monotonic semigroup is M2,1 + identity or the index two aperiodic monogenic semigroup with an identity element adjoined to it.

We can learn about this semigroup by comparing it to its monotonic commutative aperiodic counterpart. The only difference is that 2*3=1 rather then 2 this means that the middle and minimal elements can either produce their least upper bound which is the middle element or the maximal element. Since this produces a greater upper bound then it would otherwise would among comparable elements I call this chain non-monotonic. This special T3 semigroup often appears in larger partially ordered semigroups as a subsemigroup.