The dichotomy of Part(A)

The lattice $Part(A)$ is associated with two very different semilattices. One of them is semimodular and the other is not. The semimodularity condition $a \wedge b \le: a \implies a :\le a \vee b$ only really effects the semilattice used in the right hand side of the implication. We can say that $\vee$ is semimodular while $\wedge$ is not.

In fact $\vee$ has a stronger property: it is additive. The height of an element, which is the maximal chain length of its principal ideal, corresponds by semimodularity to the minimal number of atoms needed to express an element. The minimal number of atoms needed to express an element is clearly bounded by addition. So $\vee$ is additive, while $\wedge$ is multiplicative. This simplifies the dichotomy:

• $(Part(A),\vee)$ is additive
• $(Part(A),\wedge)$ is multiplicative

The lattice $Part(A)$ apparently contains both commutative arithemic operations in itself. If we take $Part(A)$ over any finite set $A$ then we see another dichotomy: $\wedge$ has exponentially more irreducibles then $\vee$. Apparently, the greater number of atoms of $\wedge$ gives it greater room to be multiplicative, while the smaller number for $\vee$ can only be added together.

The arithmetical properties of $Part(A)$ are a major part of the motivation for my use of commutative semigroup theory in the study of semilattices. Semilattices like $\wedge$ are associated with commutative semigroups, like in this case multiplication. The multiplicative property of $\wedge$ is responsible for the definition of the product in the topos of sets.

References:
[1] Inclusion-exclusion principle for set partitions
https://lisp-ai.blogspot.com/2020/11/inclusion-exclusion-principle-for-set.html