Total relation

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 xRy }. Conversely, R is called right total if Y equals the range {y : there is an x with xRy }.

When f: X → Y is a function, the domain of f is all of X, hence f is a total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f is not a total relation.

"A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."^[1]

Algebraic characterization

Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let $X, Y$ be two sets, and let $R \subseteq X \times Y .$ For any two sets $A, B,$ let $L_{A, B} = A \times B$ be the universal relation between $A$ and $B,$ and let $I_{A} = {(a, a) : a \in A}$ be the identity relation on $A .$ We use the notation $R^{⊤}$ for the converse relation of $R .$

$R$ is total iff for any set $W$ and any $S \subseteq W \times X,$ $S \neq \emptyset$ implies $S R \neq \emptyset .$ ^[2]^: 54
$R$ is total iff $I_{X} \subseteq R R^{⊤} .$ ^[2]^: 54
If $R$ is total, then $L_{X, Y} = R L_{Y, Y} .$ The converse is true if $Y \neq \emptyset .$ ^{[note 1]}
If $R$ is total, then $\overline{R L_{Y, Y}} = \emptyset .$ The converse is true if $Y \neq \emptyset .$ ^{[note 2]}^[2]^: 63
If $R$ is total, then $\overline{R} \subseteq R \overline{I_{Y}} .$ The converse is true if $Y \neq \emptyset .$ ^[2]^: 54^[3]
More generally, if $R$ is total, then for any set $Z$ and any $S \subseteq Y \times Z,$ $\overline{R S} \subseteq R \overline{S} .$ The converse is true if $Y \neq \emptyset .$ ^{[note 3]}^[2]^: 57

Notes

^ If $Y = \emptyset \neq X,$ then $R$ will be not total.
^ Observe $\overline{R L_{Y, Y}} = \emptyset \Leftrightarrow R L_{Y, Y} = L_{X, Y},$ and apply the previous bullet.
^ Take $Z = Y, S = I_{Y}$ and appeal to the previous bullet.

References

^ Functions from Carnegie Mellon University
^ ^a ^b ^c ^d ^e Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Definition 5.8, page 57.

Gunther Schmidt & Michael Winter (2018) Relational Topology
C. Brink, W. Kahl, and G. Schmidt (1997) Relational Methods in Computer Science, Advances in Computer Science, page 5, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
Gunther Schmidt & Thomas Strohlein (2012)[1987] Relations and Graphs, p. 54, at Google Books
Gunther Schmidt (2011) Relational Mathematics, p. 57, at Google Books

[3] If $Y = \emptyset \neq X,$ then $R$ will be not total.

[4] Observe $\overline{R L_{Y, Y}} = \emptyset \Leftrightarrow R L_{Y, Y} = L_{X, Y},$ and apply the previous bullet.

[6] Take $Z = Y, S = I_{Y}$ and appeal to the previous bullet.

[1] Functions from Carnegie Mellon University

[R&G-2] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[GS11-5] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). Definition 5.8, page 57.

[1]

[2]

[note 1]

[note 2]

[3]

[note 3]

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Total relation

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Algebraic characterization

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Total relation

Algebraic characterization

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Notes

References

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