Epimorphisms on category of (finite) bounded distributive lattices = surjective?

Question

So, I'm not very used to category theory language. Reading a paper now that is talking about lifts, and it uses the term epimorphism between bounded distributive lattices. Wikipedia says sometimes these are the surjective homomorphisms, sometimes not...is this a case where the epis are surjective? Does it end up being a synonym for surjective, and if not, is one a subset (category/whatever) of the other? Does it matter if the category is restricted to finite distributive lattices?

Answer 1

If by “epimorphism” you mean “right cancellable morphisms”, then it is not true that in the category of (finite) distributive lattices all epimorphisms are surjective (it is of course true that every surjective morphism is right cancellable).

For example, consider the standard diamond lattice $L=\{\mathbf{0},\mathbf{1},x,y\}$ with $x\vee y = \mathbf{1}$, and $x\wedge y = \mathbf{0}$. Let $L$ be the sublattice $M=\{\mathbf{0},x,\mathbf{1}\}$. I claim that the embedding $M\hookrightarrow L$ is an epimorphism. Indeed, let $N$ be a distributive lattice, let $f,g\colon L\to N$ be a lattice morphism, and assume that $f|_M=g|_M$. If $f(y)\neq g(y)$, then $f(y)$ and $g(y)$, are both relative complements of $f(x)=g(x)$. This is impossible in a distributive lattice, hence $f(y)=g(y)$. Thus, $f=g$; that is, any two maps morphisms from $L$ that agree on $M$ must be equal, so the embedding $M\hookrightarrow L$ is a nonsurjective epimorphism.

But you should check with the author of the paper you are reading. Too many authors use “epimorphism” as a synonym for “surjective morphism”, even in situations/contexts where epimorphism is not equivalent to surjective (e.g., in the categories of rings, or unital rings, or monoids).