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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?

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    $\begingroup$ Example of another useful relation-with-abstract-property: you could see a function as a relation where for each x, there is a unique y for which (x,y) belongs to the relation. $\endgroup$ – frabala Apr 18 '19 at 10:36
  • $\begingroup$ Since functions are relations, all kinds of useful (or not-so-useful) properties of functions you can think of can be thought of as "types of relation". $\endgroup$ – hmakholm left over Monica Apr 18 '19 at 17:13
  • $\begingroup$ @HenningMakholm. I was not thinking specially of functions. I thought of relations in general, and of all possible combinations of properties that could be imagined ( say - totally at random - connectedness+asymetry+reflexivity). I know that some combinations give (1) equiv. rel (2) orderings. I wanted to know about entirely different types, based on totally different properties and combitaions of properties. $\endgroup$ – Saint James Apr 18 '19 at 17:16
  • $\begingroup$ @EleonoreSaintJames: If you want to know "entirely different types" of classifications, yet somehow "being a function" doesn't count, then I don't understand what would count at all. $\endgroup$ – hmakholm left over Monica Apr 18 '19 at 17:25
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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.

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  • $\begingroup$ @RoddyMacPhee.Thanks a lot! I did not know this wiki page! $\endgroup$ – Saint James Apr 18 '19 at 16:56
  • $\begingroup$ the table itself is on most ordering relation pages at last check. Though they may not all have special names or be possible in reality (properties contradict each other etc.), by set theory itself, you can say that there are roughly $2^n$ relation types where n are the total number of possible properties. $\endgroup$ – Roddy MacPhee Apr 18 '19 at 16:59
  • $\begingroup$ @RoddyMacPhee.I did not check these pages before asking the question since I expected totally different kinds of relations ( I mean types of relations being neither orderings nor equivalence relation). $\endgroup$ – Saint James Apr 18 '19 at 17:02
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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.

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  • $\begingroup$ removing cycles also renames it to Directed Acyclic Graphs. $\endgroup$ – Roddy MacPhee Apr 18 '19 at 17:28
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1Yes.   Preorders, reflexive and transitive. 

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