In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty...

a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order...

i<j.} Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. (Here...

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...

well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded...

whose value is monotonically decreased with respect to a (strict) well-founded relation by the iteration of a while loop under some invariant conditions...

In mathematics, a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle...

transfinite recursion on any well-founded relation R. (R need not even be a set; it can be a proper class, provided it is a set-like relation; i.e. for any x, the...

and reflexive relation on X {\displaystyle X} ) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that...

binary relation associates some elements of one set called the domain with some elements of another set called the codomain. Precisely, a binary relation over...

mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant to suggest...

mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in...

reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to...

which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies...

either relation (union), removing tuples from the first relation found in the second relation (difference), extending the tuples of the first relation with...

pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set...

In mathematics, an asymmetric relation is a binary relation R {\displaystyle R} on a set X {\displaystyle X} where for all a , b ∈ X , {\displaystyle...

the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive. For finite sets...

{\displaystyle \in } -well-founded. For a binary relation R D {\displaystyle R_{D}} on a set D {\displaystyle D} , well-foundedness can be defined by requiring...

ccc Knaster's condition, sometimes denoted property (K) Well-founded relation Ordinal number Well-quasi-ordering Semilattice Lattice (Directed) complete...

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if: ∀ a , b ∈ X ( a R b ⇔ b R a ) , {\displaystyle...

from the domain of a well-founded relation, such as from the ordinal numbers. If the measure "decreases" according to the relation along every possible...

corresponding absorption laws. A set S partially ordered by the binary relation ≤ is a meet-semilattice if For all elements x and y of S, the greatest...

(strictly partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one...

few of the most common relations found in the literature are given below. Position–linear momentum uncertainty relation: for the position and linear momentum...

In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with...

also have to show that the loop terminates. For this we define a well-founded relation on the state space denoted as (wfs, <) and define a variant function...

the latter (→) is moreover well-founded, it is called a reduction ordering, or a reduction preorder. Given a binary relation R, its rewrite closure is...

b=a\vee b} and dually for the other direction. One can now check that the relation ≤ {\displaystyle \leq } introduced in this way defines a partial ordering...

decreases with respect to a well-founded relation < on some domain set D during each iteration. Since < is well-founded, a strictly decreasing chain...