Algebras for the covariant power-set monad are posets with all small suprema and their morphisms preserve every small supremum

Given an algebra $a$, define $x\le y$ when $a(\{x,y\})=y$. The singleton law gives reflexivity; symmetry of the set $\{x,y\}$ gives antisymmetry; and the flattening law makes the induced binary join associative, from which transitivity follows. Thus this is a partial order.

For every $A\subseteq X$, put $\sup A=a(A)$. Applying the flattening law to families formed from $A$, its singleton subsets, and an arbitrary common upper bound proves respectively that every member of $A$ lies below $a(A)$ and that $a(A)$ lies below every upper bound; this includes $A=\mathrm{\varnothing }$. The algebra-homomorphism equation $fa=b\mathcal{P}(f)$ says exactly that $f(\sup A)=\sup f[A]$.

Conversely, on a poset with all small suprema define $a(A)=\sup A$. Singleton suprema give the unit law, and $\sup (\bigcup \mathcal{A})=\sup \{\sup A:A\in \mathcal{A}\}$ gives the multiplication law. A map satisfies the algebra-homomorphism equation exactly when it preserves these suprema.