Non-commutative version of the exceptional commutative semigroup

The exceptional small commutative semigroup of order four was defined primarily by the different product of its incomparable minimal elements. Rather then then producing the least upper bound it produced the maximal element of the partial order. Amongst the non-commutative semigroups there is a similar semigroup that produces either the maximum or the join depending upon the argument order.

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 2, 2 ] ]

So in total for the same indices and the same factorisation partial order there are three different types of behavior: the semilattice like behavior, the maximizing behavior, and the argument dependent half way case. All of these semigroups, just like the T3 special semigroup on three elements discussed previously are J-trivial. So we will be considering J-trivial semigroups which are associative bound-producing functions on partial orders.