Studying lattices, I'm looking for the following construction:
Given a finite (hence bounded and complete) lattice $L$ and $a,b \in L$ such that $a\leq b$, obtain a lattice $L'$ by "contracting" the segment $[a,b]$ into one point.
There should be a homomorphism frrom $L$ to $L'$. I believe this can be describe by the equivalence classes induced by $$ x \sim \begin{cases} & x \lor b \text{ if } a \leq x\\ & x \land a \text{ if } x \leq b\\ & x \lor \neg a \text{ if } \neg b \leq x\\ & x \land \neg b \text{ if } x \leq \neg a \end{cases} $$ But showing this is well defined seems rather inelegant, especially since in my casew the lattice is complemented and I have twice as many cases. I also believe this is the result one obtains from taking the presentation of $L$ (all inequalities) and adding $a=b$ to it. This obviously induces an homomorphism to the free algebra induced by the presentation, but then I have no idea how to show that the resulting lattice is non trivial ($1\neq 0$).
It seems that this "contraction" construction would have to be standard, some kind of inverse Day's doubling procedure, but I've found no reference about it.