For Boolean algebras, the non-triviality or non-degeneracy assumption is that the zero is distinct from the one so for powerset BAs, that means taking 1 as the smallest set since it has two subsets. For partition algebras in [partition logic](http://www.ellerman.org/introduction-to-partition-logic/), the analogous non-degeneracy assumption is that the underlying set \\(X\\) has at least two elements so the indiscrete partition (one block containing both elements) is distinct from the discrete partition (which distinguishes the two elements). The interesting thing is that the Boolean logic of subsets and the logic of partitions _agree_ in this minimal case. That is, the powerset BA on the one-element set \\(\wp(1)\\) using subset operations is isomorphic to the partition algebra \\(\prod(2)\\) using operations (e.g., join, meet, and implication) on partitions.
This isomorphism seems to be the key to understanding what G. Spencer Brown was trying to do in his somewhat cryptic book _Laws of Form_. He develops an algebra based on intuitive reasoning about "the Distinction" (e.g., distinguishing the two elements in \\(2\\) but ends up with the two-element Boolean algebra as many commentators have pointed out.