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Transitive binary relations
v
t
e
Symmetric
Antisymmetric
Connected
Well-founded
Has joins
Has meets
Reflexive
Irreflexive
Asymmetric
Total, Semiconnex
Anti- reflexive
Equivalence relation
Y
✗
✗
✗
✗
✗
Y
✗
✗
Preorder (Quasiorder)
✗
✗
✗
✗
✗
✗
Y
✗
✗
Partial order
✗
Y
✗
✗
✗
✗
Y
✗
✗
Total preorder
✗
✗
Y
✗
✗
✗
Y
✗
✗
Total order
✗
Y
Y
✗
✗
✗
Y
✗
✗
Prewellordering
✗
✗
Y
Y
✗
✗
Y
✗
✗
Well-quasi-ordering
✗
✗
✗
Y
✗
✗
Y
✗
✗
Well-ordering
✗
Y
Y
Y
✗
✗
Y
✗
✗
Lattice
✗
Y
✗
✗
Y
Y
Y
✗
✗
Join-semilattice
✗
Y
✗
✗
Y
✗
Y
✗
✗
Meet-semilattice
✗
Y
✗
✗
✗
Y
Y
✗
✗
Strict partial order
✗
Y
✗
✗
✗
✗
✗
Y
Y
Strict weak order
✗
Y
✗
✗
✗
✗
✗
Y
Y
Strict total order
✗
Y
Y
✗
✗
✗
✗
Y
Y
Symmetric
Antisymmetric
Connected
Well-founded
Has joins
Has meets
Reflexive
Irreflexive
Asymmetric
Definitions, for all and
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation be transitive: for all if and then
A term's definition may require additional properties that are not listed in this table.
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 one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation).
Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order.
A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain).
and 26 Related for: Connected relation information
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relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness...
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the European Union, in response to a complaint about the restrictions in relation to an antitrust case involving Microsoft, ruled that "so far, there are...
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...
the way train cars are connected on a track. When a reader or listener "loses the train of thought" (i.e., loses the relation between consecutive sentences...
pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set...
structurally connected to one another. A widely upheld position is that in coherent discourse, every individual utterance is connected by a discourse relation with...
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...
Partial order that arises as the subset-inclusion relation on some collection of objects Region – Connected open subset of a topological spacePages displaying...