relation_is_Ferrers(x)
relation_is_antisymmetric(x)
relation_is_asymmetric(x)
relation_is_bijective(x)
relation_is_binary(x)
relation_is_complete(x)
relation_is_coreflexive(x)
relation_is_crisp(x)
relation_is_endorelation(x)
relation_is_equivalence(x)
relation_is_functional(x)
relation_is_homogeneous(x)
relation_is_injective(x)
relation_is_interval_order(x)
relation_is_irreflexive(x)
relation_is_left_total(x)
relation_is_linear_order(x)
relation_is_match
relation_is_negatively_transitive
relation_is_partial_order(x)
relation_is_reflexive(x)
relation_is_right_total(x)
relation_is_semiorder(x)
relation_is_semitransitive(x)
relation_is_strict_linear_order(x)
relation_is_strict_partial_order(x)
relation_is_strongly_complete(x)
relation_is_surjective(x)
relation_is_symmetric(x)
relation_is_tournament(x)
relation_is_transitive(x)
relation_is_weak_order(x)
relation_is_preference(x)
relation_is_preorder(x)
relation_is_quasiorder(x)relation.A relation $(R) on a set \eqn{X} is called \emph{homogeneous} iff \eqn{D(R) = (X, \dots, X)}
An \emph{endorelation} is a binary homogeneous relation.
For a crisp binary relation, let us write \eqn{x R y} iff \eqn{(x, y)} is contained in \eqn{R}.
A crisp binary relation \eqn{R} is called \describe{ \item{left-total:}{for all \eqn{x} there is at least one \eqn{y} such that \eqn{x R y}.} \item{right-total:}{for all \eqn{y} there is at least one \eqn{x} such that \eqn{x R y}.} \item{functional:}{for all \eqn{x} there is at most one \eqn{y} such that \eqn{x R y}.} \item{surjective:}{the same as right-total.} \item{injective:}{for all \eqn{y} there is at most one \eqn{x} such that \eqn{x R y}.} \item{bijective:}{left-total, right-total, functional and injective.} }
A crisp endorelation \eqn{R} is called \describe{ \item{reflexive:}{\eqn{x R x} for all \eqn{x}.} \item{irreflexive:}{there is no \eqn{x} such that \eqn{x R x}.} \item{coreflexive:}{\eqn{x R y} implies \eqn{x = y}.} \item{symmetric:}{\eqn{x R y} implies \eqn{y R x}.} \item{asymmetric:}{\eqn{x R y} implies that not \eqn{y R x}.} \item{antisymmetric:}{\eqn{x R y} and \eqn{y R x} imply that \eqn{x = y}.} \item{transitive:}{\eqn{x R y} and \eqn{y R z} imply that \eqn{x R z}.} \item{complete:}{for all distinct \eqn{x} and \eqn{y}, \eqn{x R y} or \eqn{y R x}.} \item{strongly complete}{for all \eqn{x} and \eqn{y}, \eqn{x R y} or \eqn{y R x} (i.e., complete and reflexive).} \item{negatively transitive}{not \eqn{x R y} and not \eqn{y R z} imply that not \eqn{x R z}.} \item{Ferrers}{\eqn{x R y} and \eqn{z R w} imply \eqn{x R w} or \eqn{y R z}.} \item{semitransitive}{\eqn{x R y} and \eqn{y R z} imply \eqn{x R w} or \eqn{w R z}.} }
Some combinations of these basic properties have special names because of their widespread use: \describe{ \item{preorder:}{reflexive and transitive.} \item{quasiorder:}{the same as preorder.} \item{equivalence:}{a symmetric preorder (reflexive, symmetric and transitive).} \item{weak order:}{complete and transitive.} \item{preference:}{the same as weak order.} \item{partial order:}{antisymmetric and transitive.} \item{strict partial order:}{an irreflexive partial order (i.e., irreflexive, antisymmetric and transitive, or equivalently: asymmetric and transitive).} \item{linear order:}{a complete partial order.} \item{match:}{strongly complete.} \item{strict linear order:}{an irreflexive linear order, or equivalently: a complete strict partial order.} \item{tournament:}{complete and asymmetric.} \item{interval order:}{complete and Ferrers.} \item{semiorder:}{a semitransitive interval order.} }
If \eqn{R} is a weak order (\dQuote{weak preference relation}), \eqn{I = I(R)} defined by \eqn{x I y} iff \eqn{x R y} and \eqn{y R x} is an equivalence, the \emph{indifference relation} corresponding to \eqn{R}.
There seem to be no commonly agreed definitions for order relations: e.g., Fishburn (1972) requires these to be irreflexive.
For a fuzzy binary relation \eqn{R}, let \eqn{R(x, y)} denote the
membership of \eqn{(x, y)} in the relation. Write \eqn{T} and \eqn{S}
for the fuzzy t-norm (intersection) and t-conorm (disjunction),
respectively (min and max for the \dQuote{standard} Zadeh family).
Then generalizations of the above basic endorelation predicates are as
follows.
\describe{
\item{reflexive:}{\eqn{R(x, x) = 1} for all \eqn{x}.}
\item{irreflexive:}{\eqn{R(x, x) = 0} for all \eqn{x}.}
\item{coreflexive:}{\eqn{R(x, y) > 0} implies \eqn{x = y}.}
\item{symmetric:}{\eqn{R(x, y) = R(y, x)} for all \eqn{x, y}.}
\item{asymmetric:}{
\eqn{T(R(x, y), R(y, x)) = 0} for all \eqn{x, y}.}
\item{antisymmetric:}{
\eqn{T(R(x, y), R(y, x)) = 0} for all \eqn{x \ne y}.}
\item{transitive:}{\eqn{T(R(x, y), R(y, z)) \le R(x, z)} for all
\eqn{x, y, z}.}
\item{complete:}{
\eqn{S(R(x, y), R(y, x)) = 1} for all \eqn{x \ne y}.}
\item{strongly complete:}{
\eqn{S(R(x, y), R(y, x)) = 1} for all \eqn{x, y}.}
\item{negatively transitive:}{
\eqn{R(x, z) \le S(R(x, y), R(y, z))} for all \eqn{x, y, z}.}
\item{Ferrers:}{
\eqn{T(R(x, y), R(z, w)) \le S(R(x, w), R(z, y))} for all
\eqn{x, y, z, w}.}
\item{semitransitive:}{
\eqn{T(R(x, w), R(w, y)) \le S(R(x, z), R(z, y))} for all
\eqn{x, y, z, w}.}
}
The combined predicates are obtained by combining the basic predicates
as for crisp endorelations (see above).$
H. R. Varian (2002),
Intermediate Microeconomics: A Modern Approach.
6th Edition. W. W. Norton & Company.