# wikipedia relation algebra example #3

Wikipedia article gives the following relation algebra example:

3 An important generalization of the previous example is the power set $2^E$ where E ⊆ X² is any equivalence relation on the set X. This is a generalization because X² is itself an equivalence relation, namely the complete relation consisting of all pairs. While $2^E$ is not a subalgebra of $2^{X²}$ when E ≠ X² (since in that case it does not contain the relation X², the top element 1 being E instead of X²), it is nevertheless turned into a relation algebra using the same definitions of the operations. Its importance resides in the definition of a representable relation algebra as any relation algebra isomorphic to a subalgebra of the relation algebra $2^E$ for some equivalence relation E on some set. The previous section says more about the relevant metamathematics.

I fail to understand it, can somebody please give an example, perhaps similar to how one would describe algebra of binary relations in example #2?

I can see that equivalence relations are partitions of a set which have lattice structure (with join being partition intersection and union being transitive closure). Equivalence relations essentially borrow lattice structure from more general algebra of binary relations. However, neither composition, nor complement respect equivalence constraint, therefore equivalence relations can't be elements of relation algebra.

-