# Generalization of binary relation properties to ternary relations

The properties of binary relations such as reflexivity, symmetry, transitivity are ubiquitous. What about ternary relations, which properties can we consider fundamental? For motivation here is the one that looks like generalized transitivity:

R(e,a,b) & R(d,e,c) & R(f,b,c) -> R(d,a,f)


This is a well known algebraic law disguised in relational notation (which one?-).

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In general algebra, relations of finite arity are sometimes viewed as subuniverses of a finite power of some universe A. One can build functions which respect one or more relations and thus get a universal algebra. Then properties such as diagonal subalgebras, subdirectly irreducible algebras, and the like are studied. There are also relation algebras which show how to form other binary relations on a set from given ones; I do not know about higher arity analogues of such but that might be something else to search. Gerhard "Email Me About System Design" Paseman, 2011.06.22 –  Gerhard Paseman Jun 23 '11 at 0:58
Also, my guess is associatvity. Gerhard "Overdosed On Associativity At Berkeley" Paseman, 2011.06.22 –  Gerhard Paseman Jun 23 '11 at 1:04
If $e=(a+b)$, $d=e+c$, $f=b+c$, then $d=a+f$ by associativity, since $(a+b)+c=a+(b+c)$. –  Joel David Hamkins Jun 23 '11 at 1:51
Associativity, correct:-) Assuming that relation R is a function, I hope that my implication is equivalent to associativity, not just follows from it. –  Tegiri Nenashi Jun 23 '11 at 2:06
Yes, that is what I meant; assuming your relation is expressing the graph of a binary function, it exactly expresses associativity. –  Joel David Hamkins Jun 23 '11 at 2:10