I'm a beginning student of category theory so the question is a little hazy. Apologies if it is too basic.
An equivalence relation induces a "symmetric category" (bad terminology?), where you can back from any arrow. The category induced by a group has a different symmetry. How are these two specifically related? Is an equivalence relation somehow an algebra, like a group, that specializes the category axioms? Is it more deeply analogous to a group in some way?
I know that a category can also be induced by a partial order - which encodes anti-symmetry rather than symmetry . Is there a corresponding algebra encoding antisymetry (like a group but encoding anti-symmetry instead)? I know a partial order itself has the algebra of a lattice.
A set with an equivalence relation is often called a setoid. Categorically, a setoid is a thin groupoid. A groupoid may be thought of as a "multi-object group" in the same way that a category is a "multi-object monoid": that is, the endomorphisms of every object in a groupoid form a group.
A partial order is a thin skeletal category (a preorder is simply a thin category). Therefore, the algebraic structure corresponding to a partial order (or preorder), in the same way that groups correspond to equivalence relations, is a monoid.
The relationship "an X is just a one-object Y" is called horizontal categorification, where for your examples we have: