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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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

inform us that the measure (COUNT X) decreases according to the well-founded relation LESSP in each induction step of the scheme. The above induction...

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

binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of'...

In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all distinct pairs of elements of the set in...

{\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...

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...

\wedge )} is then a meet-semilattice. Moreover, we then may define a binary relation ≤ {\displaystyle \,\leq \,} on A, by stating that x ≤ y {\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...

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

mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are...

this simulated set theory follows from E's extensionality. From its well-foundedness follows an axiom of foundation. There remains the question of what...

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