The exceptional small commutative aperiodic semigroup

Commutative aperiodic semigroups can be studied in order to consider generalizations of semilattices, and to determine their lattice like properties. Ultimately, we can define order-preserving commutative aperiodic semigroups by the joins of distinct irreducible elements are preserved under the algebraic preordering. It is clear that this is the case for all commutative aperiodic semigroups of size three or less, but in the case of aperiodic semigroups of size four it is true for 30/31 of them, all except for a single exceptional case. This semigroup is identified as the GAP small semigroup 4, 34 and it can be considered the simplest counter-case to the order-preserving property.

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

Every single semigroup is associated with a preordering relation, and for commutative aperiodic semigroups this is a partial ordering relation. In the case of this commutative aperiodic semigroup, its algebraic ordering is the weak order [2 1 1], which is shown below.



The main issue with this small commutative aperiodic semigroup is that the product of 3 and 4 goes to 1 rather then to 2 which is the least upper bound, or the join, of the algebraic preordering. As a result, it does not preserve the ordering of its elements in its operation. It is clear that this is the unique smallest commutative aperiodic semigroup that has this property because it must have the algebraic preordering [2 1 1] so that distinct minimal elements can go to an upper bound which is different from the least one and this is the only such semigroup that does. This principle allows us to compare commutative aperiodic semigroups to semilattices in order to better generalize the properties of lattice theory to other commutative structures.