# What are products and coproducts of bounded lattices?

I understand the universal property that products and coproducts must satisfy, but I'm having trouble finding a concrete construction in the category Lat of bounded lattices with meet/join top/bottom preserving functions as arrows.

For example, does Lat have products and coproducts? What lattice would constitute the co/product of simple lattices like:

$$0\rightarrow a \rightarrow 1 \quad\text{and}\quad 0\rightarrow b \rightarrow 1$$

and what about more complicated lattices?

I suspect that one construct may be something like $\{ \langle a, b\rangle : a \in L_1, b\in L_2\}$ where we define $\langle a_1, b_1\rangle \preceq \langle a_2, b_2\rangle$ in $L_1\times L_2$ if and only if $a_1 \leq a_2$ and $b_1 \leq b_2$. The new top and bottom elements would then be $\langle 1, 1\rangle$ and $\langle 0,0\rangle$. If this is the coproduct, the inclusions might be $a\mapsto \langle a, 1\rangle$, for example. If this is the product, the projections might be $\langle a, b\rangle \mapsto a$ as in ordinary set projection.

Bounded lattices can be described by an equational theory, so they are examples of models of a Lawvere theory. Thus the category of bounded lattices is complete and cocomplete. Furthermore, the underlying set of the product is the product of the underlying sets. Basically, it will be a pair of bounded lattices with operations defined component-wise. In fact, this will be more or less the situation for all limits and as well as sifted colimits. So the product has elements $\{(0,0),(1,1),(a,0),(a,b),(a,1),(0,b),(1,b)\}$ where e.g. $(x_1,y_1)\land(x_2,y_2)=(x_1\land x_2,y_1\land y_2)$.
Your examples are free bounded lattices generated by singleton sets, so the coproduct is going to be the free bounded lattice generated by a two element set. In particular, it will have elements $\{0,1,a,b,a\land b,a\lor b\}$.
• @user326210 Limits are straightforward. Just take the limit of the underlying sets and equip it with the (restricted) (product of) the original operation(s). The pullback of $f:A\to C$ along $g:B\to C$ of two lattice homomorphisms is the set $\{(a,b)\in A\times B\mid f(a)=g(b)\}$ equipped with the original lattice operations applied component-wise and restricted. Because $f$ and $g$ are lattice homomorphisms if $f(a_i)=g(b_i)$ then $f(a_1\land a_2)=g(b_1\land b_2)$ and similarly for $\lor$, so the component-wise operations will be closed on the subset the pullback picks out. – Derek Elkins Jul 3 '17 at 3:06