- antisymmetric
- \(a R b \land b R a \rightarrow a = b\)
- automorphic
- endomorphic and isomorphic
- bijective
- injective and surjective
- binary relation
- a binary relation between sets \(A\) and \(B\) is a subset of \(A \times B\)
- connex
- \(a R b \lor b R a\)
- epimorphism
- right-cancellative morphism
- equivalence relation
- symmetric preorder
- function
- binary relation that is univalent and left-total
- idempotent
- \(a^2 = a\)
- involutive
- \(a^2 = 1\)
- injective
- \(f(a) = f(b) \rightarrow a = b\)
- left-total
- \(\forall a \in A: \exists b \in B: a R b\)
- monomorphism
- left-cancellative morphism
- monotone
- \(a \leq b \rightarrow f(a) \leq f(b)\)
- partial order
- antisymmetric preorder
- prefix order
- downward-total partial order
- preorder
- binary relation that is reflexive and transitive
- reflexive
- \(a R a\)
- strictly monotone
- \(a < b \rightarrow f(a) < f(b)\)
- surjective
- \(\text{im}(f) = \text{cod}(f)\)
- symmetric
- \(a R b \rightarrow b R a\)
- transitive
- \(a R b \land b R c \rightarrow a R c\)
- univalent
- \(a R b \land a R c \rightarrow b = c\)