Intransitivity
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stro
Euclidean relation
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."
Equality (mathematics)
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
Ancestral relation
In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation R is its transitive closure, however defined in a different way, see below. Ancestral relations make t
Symmetric relation
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 i
Better-quasi-ordering
In order theory a better-quasi-ordering or bqo is a quasi-ordering that does not admit a certain type of bad array. Every better-quasi-ordering is a well-quasi-ordering.
Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A po
Join and meet
In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum (least upper bound) of denoted and similarly, the meet of is the infimum (greatest lower boun
Well-quasi-ordering
In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements from contains an increasing pair with
Converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relat
Separoid
In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical order induced by inclusion. Many mathematical objects which appear to be quite differ
Rational consequence relation
In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listed below.
Asymmetric relation
In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to
Covering relation
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering rel
Quotient by an equivalence relation
In mathematics, given a category C, a quotient of an object X by an equivalence relation is a coequalizer for the pair of maps where R is an object in C and "f is an equivalence relation" means that,
Total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and
Category of relations
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A ×
Prewellordering
In set theory, a prewellordering on a set is a preorder on (a transitive and strongly connected relation on ) that is wellfounded in the sense that the relation is wellfounded. If is a prewellordering
Transitive closure
In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets, "smallest" can be taken in its usual sense, o
Semiorder
In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores with
Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in
Connected relation
In mathematics, a relation on a set is called connected or total if it relates (or "compares") all distinct pairs of elements of the set in one direction or the other while it is called strongly conne
Directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set together with a reflexive and transitive binary relation (that is, a preorder), with the additional property
Equipollence (geometry)
In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB from point A to point B has the opposite direction to line segment BA. Two parallel line segm
Idempotent relation
In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. This notion general
Homogeneous relation
In mathematics, a homogeneous relation (also called endorelation) over a set X is a binary relation over X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "
Weak ordering
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 a
Binary relation
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 pair
Transitive relation
In mathematics, a 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. Each partial order as well as each equivalenc
Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set
Equivalence class
In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence cla
Preorder
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) parti
Accessibility relation
An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth valu
Reflexive relation
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, sin
Symmetric closure
In mathematics, the symmetric closure of a binary relation on a set is the smallest symmetric relation on that contains For example, if is a set of airports and means "there is a direct flight from ai
Dependency relation
In computer science, in particular in concurrency theory, a dependency relation is a binary relation on a finite domain , symmetric, and reflexive; i.e. a finite tolerance relation. That is, it is a f
Reflexive closure
In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R. For example, if X is a set of distinct numbers and x R y means "x is le
Series-parallel partial order
In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations. The series-paralle
Dense order
In mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is
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 equivale
Law of trichotomy
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero. More generally, a binary relation R on a set X is trichotomous if for all x and y in X, exact
Serial relation
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers
Well-founded relation
In mathematics, a binary relation R is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m not related by s
Foundational relation
In set theory, a foundational relation on a set or proper class lets each nonempty subset admit a relational minimal element. Formally, let (A, R) be a binary relation structure, where A is a class (s
Antisymmetric relation
In mathematics, a binary relation on a set is antisymmetric if there is no pair of distinct elements of each of which is related by to the other. More formally, is antisymmetric precisely if for all o
Dependence relation
In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let be a set. A (binary) relation between an element of and a subset of is called a depe
Comparability
In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not co
Quasitransitive relation
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