I am wondering about generalisations of the concept of equivalence relations to suplattices.


Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices there is a tensor product, giving $\mathcal{P}(X)\otimes\mathcal{P}(X)=\mathcal{P}(X\times X)$. Now we can define the diagonal $\Delta:=\left\{(x,x)\mid x\in X\right\}\in\mathcal{P}(X\times X)$, which is an equivalence relation. Take a general suplattice $M$. There is a tensor product $M\otimes M$, which can be embedded into the suplattice $\mathcal{P}(M\times M)$ ([reference](http://www.tac.mta.ca/tac/volumes/24/11/24-11.ps)). The elements of $\mathcal{P}(M\times M)$ can be projected to the first (or to the second) component and you can take the supremum of the resulting subset of $M$. Thus you get projections $\pi\_0,\pi\_1\colon M\otimes M\to M$, preserving suprema.


In this context you could call a $\Delta\in M\otimes M$ an “equivalence relation in $M$ if it satisfies these conditions:


* Symmetry: For all $a,b\in M$ such that $a\otimes b\le\Delta$ we have $b\otimes a\le\Delta$.
* Transitivity: For all $a,b,c\in M$ such that $a\otimes b\le\Delta$ and $b\otimes c\le\Delta$ we have $a\otimes c\le\Delta$.
* Reflexivity: The projections of the relation are maximal: $\pi\_0(\Delta)=\pi\_1(\Delta)=\top$ where $\top\in M$ is the top of the lattice.


Clearly, the diagonal is an equivalence relation in $\mathcal{P}(X)$. However, 


I am interested in ways to construct such relations in non-trivial cases (i. e. where the lattice is not the powerset of a set and $\Delta$ should not be the top itself). Is there such a concept in the literature? Has it been studied? Or do you have any comment? Non-trivial examples (maybe there are even canonical examples?) or objections regarding the soundness of the definition are welcome.


My original motivation for regarding these concepts is the definition of the semantics of equality in certain logics interpreted in $\mathcal{P}(X\times\ldots\times X)$ using the diagonal—I am not an expert in the theory of suplattices etc. Best regards



Equivalence relations in suplattices
It seems like a more natural way to generalize equivalence relations from $P(X)$ to lattices is to look at partitions. One defines a partition of a lattice L(or Boolean algebra, frame, or whatever) to be a subset $p\subseteq L\setminus\{0\}$ such that $\bigvee p=1$ and where $a\wedge b=0$ whenever $a,b\in p,a\neq b$. The obvious advantage of partitions in this case would be that the definition of a partition is much simpler and it does not involve any kinds of tensor products. I personally have researched partitions on Boolean algebras and frames and I have found them quite useful and interesting since they relate Boolean algebras to diverse topics such as uniform spaces, the ultrapower construction, pro-sets, and point-free topology. A good question to ask is if there is a one-to-one correspondence between partitions on lattices and the equivalence relations as you defined them.


