I'm trying to find a good phrase to describe the following simple property of a function $f \colon A \times B \to A$, where $A$ is some set and $B$ is some partially ordered set: For all $a \in A$ and $x,y \in B$ it holds that $x \leq y$ implies $$f(f(a,x),y) = f(a,y)$$
Examples of functions satisfying this property (or the related property with the inequality reversed) are set union and intersection ($A=B$, ordered by inclusion), orthogonal projection onto some Hilbert subspace, taking conditional expectation etc.
For conditional expectation, the above property is known as the tower property and of course one could simply continue to use this description for the abstract property, but I suspect that there is a better description, maybe using vocabulary from algebra or category theory.
Just to be clear: I'm looking for a plain English phrase (formula and variable free).
Thanks in advance!
So, what you have is just a functor from A to the category Idem of Idempotent functions. Objects in Idem are idempotent functions (satisfying $f_x\circ f_x=f_x$) and there is a morphism from $f_y$ to $f_x$ if $f_y\circ f_x=f_y$. Specifically, your functor maps $x\in B$ into $f(.,x)$, and the mapping of morphisms guarantees that $f(f(.,x),y)=f(.,y)$ whenever $y\geq x$.