Ternary equivalence relation

Last updated January 19, 2024

In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. 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 among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the same way, a binary equivalence relation on a set determines a partition.

Definition

A ternary equivalence relation on a set $X$ is a relation $E \subset X 3$ , written $[a, b, c]$ , that satisfies the following axioms:

Symmetry: If $[a, b, c]$ then $[b, c, a]$ and $[c, b, a]$ . (Therefore also $[a, c, b]$ , $[b, a, c]$ , and $[c, a, b]$ .)
Reflexivity: $[a, b, b]$ . Equivalently, in the presence of symmetry, if $a$ , $b$ , and $c$ are not all distinct, then $[a, b, c]$ .
Transitivity: If $a \neq b$ and $[a, b, c]$ and $[a, b, d]$ then $[b, c, d]$ . (Therefore also $[a, c, d]$ .)

Related Research Articles

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets $X$ and $Y$ is a new set of ordered pairs $(x, y)$ consisting of elements $x$ from $X$ and $y$ from $Y$ . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element $x$ is related to an element $y$ , if and only if the pair $(x, y)$ belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case $n = 2$ of an $n$ -ary relation over sets $X 1, ..., X n$ , which is a subset of the Cartesian product

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 equality. Any number $is equal to itself (reflexive). If, then (symmetric). If and, then (transitive).$

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation.

In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between $A$ and $B$ is written $A = B$ , and pronounced " $A$ equals $B$ ". The symbol " $=$ " is called an "equals sign". Two objects that are not equal are said to be distinct.

In mathematics, a binary relation $on a set is reflexive if it relates every element of to itself.$

A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if:

In mathematics, a binary relation $R$ on a set $X$ is transitive if, for all elements $a$ , $b$ , $c$ in $X$ , whenever $R$ relates $a$ to $b$ and $b$ to $c$ , then $R$ also relates $a$ to $c$ .

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis.

In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as " $a < b$ ". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation $[a, b, c]$ , meaning "after $a$ , one reaches $b$ before $c$ ". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order.

<span class="mw-page-title-main">Weak ordering</span> Mathematical ranking of a set

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets and are in turn generalized by (strictly) partially ordered sets and preorders.

Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity. As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" and "congruence". The system contains infinitely many axioms.

In mathematics, a partial equivalence relation is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence 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.

The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social choice theory and microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept was introduced by Sen (1969) to study the consequences of Arrow's theorem.

In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as $to distinguish from the negation of equality which is weaker.$

In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."

In mathematics, a homogeneous relation on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product $X \times X$ . This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people.

In mathematics, a relation on a set may, or may not, hold between two given members of the set. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values $1$ and $3$ , and likewise between $3$ and $4$ , but not between the values $3$ and $1$ nor between $4$ and $4$ , that is, $3 < 1$ and $4 < 4$ both evaluate to false. As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not.

In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure. Thus a tolerance is like a congruence, except that the assumption of transitivity is dropped. On a set, an algebraic structure with empty family of operations, tolerance relations are simply reflexive symmetric relations. A set that possesses a tolerance relation can be described as a tolerance space. Tolerance relations provide a convenient general tool for studying indiscernibility/indistinguishability phenomena. The importance of those for mathematics had been first recognized by Poincaré.

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