A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and total.
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Instead, a cyclic order is defined as a ternary relation, meaning " after, one reaches before ".
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A ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a partial cyclic order.
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Instead, we use a ternary relation denoting that elements,, occur after each other ( not necessarily immediately ) as we go around the circle.
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One structure that weakens this axiom is a CC system : a ternary relation that is cyclic, asymmetric, and total, but generally not transitive.
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Two internal components of a " complex object " can express ( the above ) ternary relations, using the whole object as a frame of reference.
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The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.
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A ternary relation that is " asymmetric " under cyclic permutation and " symmetric " under reversal, together with appropriate versions of the transitivity and totality axioms, is called a betweenness relation.
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A " typing relation " \ Gamma \ vdash e \ ! : \ ! \ sigma indicates that e is a term of type \ sigma in context \ Gamma, and is thus a ternary relation between contexts, terms and types.
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By currying the arguments of the ternary relation, one can think of a cyclic order as a one-parameter family of binary order relations, called " cuts ", or as a two-parameter family of subsets of, called " intervals ".
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