Commutative semigroups of order [2 1 1]

Let [2,1,1] be the weak order on four elements. It has up to isomorphism three types of elements: lower elements, middle elements, and upper elements denoted L,M, and U for short. There are also four types of unordered pairs up to isomorphism: $L^2, LM, LU, ML$. We want to partially order each semigroup isomorphism type with this factorisation order pointwise up to isomorphism. In order to do this it is necessary to introduce some measure of how much larger a given output is then expected.

The measure of the greatness of an output is the covering distance from its original location. We see that given this partial order, $L+1$ means that a given minimal element iterates to produce the middle element and $L+2$ means it iterates to produce the maximal element. It is clear then that $L+2$ is greater then $L+1$ in the ordering, and so on. Given this format, we can determine where a given commutative semigroup isomorphism type is based upon a set of relations like these on isomorphism classes of subsets. So for example, this is the poset of commutative semigroup isomorphism types with factorisation order [2 1 1]:

As previously mentioned, the other upper bounded partial orders of size four or less have commutative semigroup families that either form total orders or boolean algebras. Total orders have boolean algebras of semigroup types determined by sets of idempotents, and max height two partial orders have total orders determined by the number of idempotents. Both have a unique maximal upper bound, which is the unique commutative nilpotent semigroup of that order type.

The lower bound of the above poset is the unique semilattice with that order type, which is the least upper bound. However, there are two different maximal elements which demonstrates that partial orders don't always have a greatest upper bound commutative semigroup. In this case, in order for the factorisation order to be [2,1,1] the lower elements must produce the middle element somehow and there are two ways to that: individually or together. This produces the two different maximal commutative semigroups with this factorisation order.