coproducts in the category of lattices How to describe coproducts in the category of lattices (where objects are lattices and morphisms are lattice homomorphisms)?
 A: If you are in any variety of algebras, the coproduct of algebras $L_1$ and $L_2$ can be indicated by writing down a presentation for each $L_i$, taking the union of the presentations, and the coproduct will be the object with the union presentation.
This can be made somewhat more concrete for lattices. The default presentation for a lattice $L$ is $\langle L\;|\;R\rangle$, using all elements of $L$ as generators and all relations of the form $a\vee b = c$ and $a\wedge b = d$ which hold in $L$.
So, one presentation for the coproduct of lattices $L_1$ and $L_2$ has the form $\langle L_1\uplus L_2\;|\; R_1\uplus R_2\rangle$ where $L_1\uplus L_2$ is the disjoint union of the sets $L_1$ and $L_2$, while $R_1\uplus R_2$ essentially describes a partial join and partial meet on the disjoint union $L_1\uplus L_2$. These partial operations are such that the partial join or meet of two elements lying in a single summand $L_i$ exist and are the same as if they had been computed in the summand $L_i$. The partial join or meet of an element of $L_1$ with an element of $L_2$ is not defined.
If you freely generate a lattice with the partial lattice just described, and do it in a way that preserves the partial join and meet operations, you will get the coproduct of $L_1$ and $L_2$ in the category of lattices. (Here I mean that you should freely generate a lattice with $L_1\uplus L_2$ so that the final join and meet operations extend the starting partial operations.) This free completion is the difficult step of the construction, and it is described in
Dean, R. A.
Free lattices generated by partially ordered sets and preserving bounds.
Canadian J. Math. 16 (1964), 136-148.
The free completion of a partial lattice will consist of words in the generators, and Dean's paper gives an algorithm to decide if two words are equal or comparable.
