mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 relation ...

analogous to a

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

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. Each equivalence relation ...

. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of

collinearity In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...

among three points in

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

. 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, leaving a trail ...

'' 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'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

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 ''incidenc ...

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

partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

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, 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