Types of relations , besides equivalence relation and ordering relation? In order to define types of relations, mathematicians combine abstract properties such as reflexivity, transitivity, etc. 
For example  ( after Partee, Mathematical methods in linguistics) : 


*

*equivalence = reflexivity+ symmetry+transitivity 

*week ordering = reflexivity + antisymmetry 

*strong ordering = irreflexivity + asymmetry 
A question I ask myself is: are there other mathematically usefull combinations? Are there other types of relations besides equivalence relations and ordering relations? 
 A: Copied text from wikipedia article total order:
Transitive and reflexive relations $$
\begin{array}{lcccccc}&\text{Symmetric} &\text{Antisymmetric} & \text{Connex} & \text{Well-founded} &\text{Has joins} & \text{Has meets} \\
\text{Equivalence relation}& ✓ &✗ &✗ &✗ &✗ &✗ \\
\text{Preorder (Quasiorder)}& ✗ &✗ &✗ &✗ &✗ &✗\\
\text{Partial order}& ✗ &✓ &✗ &✗ &✗ &✗\\
\text{Total preorder}& ✗ &✗ &✓ &✗ &✗ &✗\\
\text{Total order}& ✗ &✓ &✓ &✗ &✗ &✗\\
\text{Prewellordering} & ✗ &✗ &✓ &✓ &✗ &✗\\
\text{Well-quasi-ordering} &✗ &✗ &✗ &✓ &✗ &✗\\
\text{Well-ordering} &✗ &✓ &✓ &✓ &✗ &✗\\
\text{Lattice} &✗ &✓ &✗ &✗ &✓ &✓\\
\text{Join-semilattice} &✗ &✓ &✗ &✗ &✓ &✗\\
\text{Meet-semilattice} &✗ &✓ &✗ &✗ &✗ &✓\\\end{array}$$
Functions, are simply a type of relation with every input in it's input subset, related to exactly 1 output. etc. If 2 or more inputs relate via the function to the same output, the function is surjective. An operation is a function that relates a set of n-tuples(from a Cartesian Product) to another set of outputs. The relation "is connected to" comes up in graph theory. "is the multiplicative inverse of" will show up in ring-theory. etc. very few things can't be turned into set theory in higher math, that means a lot of things are relations. Even binary operation properties, are able to be put into terms of relations. Commutativity, is simply a statement of n-tuples related like (a,b) and (b,a) being related by "is equivalent without order to"  giving the same output when the operation is applied.
A: Functions are special relations, so any kind of useful (or not so useful) property of functions can be thought of as a "type of relation"; for example


*

*Injective function.

*Permutation.

*(et cetera ad libitum)


A simple graph can be viewed as an irreflexive symmetric relation and vice versa -- so all properties of graphs can be considered a "type of relation". There's a huge number of potentially interesting properties -- for example:


*

*Planar graph.

*Bipartite graph.

*Tree.

*Cubic graph.

*Connected graph.

*4-colorable graph.

*(et cetera ad nausaeam)


If you remove the symmetry requirement, you get directed graphs, with interesting properties of their own.
A: 1Yes.  
Preorders, reflexive and transitive.   
