mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a ternary equivalence relation is a kind of

ternary relation In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relatio ...

analogous to a

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

. A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below. The classic example is the relation of

collinearity In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

among three points in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or ''

pencils A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion (mechanical), abrasi ...

'' that form a

linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...

in the sense of

incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...

. In the same way, a binary equivalence relation on a set determines a partition.

Definition

A ternary equivalence relation on a set is a relation , written , that satisfies the following axioms: #Symmetry: If then and . (Therefore also , , and .) #Reflexivity: . Equivalently, in the presence of symmetry, if , , and are not all distinct, then . #Transitivity: If and and then . (Therefore also .)

References

* * * * * * * * * *{{Citation , first=Wanda , last=Szmielew, authorlink=Wanda Szmielew , title=On ''n''-ary equivalence relations and their application to geometry , publisher=Instytut Matematyczny Polskiej Akademi Nauk , year=1981 , location=Warsaw , url=http://eudml.org/doc/268578 Mathematical relations Incidence geometry Projective geometry