Heterogeneous relation

difunctionalInduced concept latticerectangular relationrelationdifunctional relationFringe of a relationleft residualrelations
In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B, where A and B are distinct sets.wikipedia
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Algebraic logic

calculus of relationslogic of relationsalgebraization
Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations.
A homogeneous binary relation is found in the power set of X × X for some set X, while a heterogeneous relation is found in the power set of X × Y, where X ≠ Y.

Bipartite graph

bipartitebipartite graphsbipartiteness
For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph.
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally.

Converse relation

converseinverseinverse relation
The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations.
Similarly, the category of heterogeneous relations, Rel is also an ordered category.

Complete bipartite graph

bicliquecomplete bipartite subgraphcomplete bipartite subgraphs
Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, they are the "concepts" that generate a lattice associated with a relation.
When a lattice is formed by taking meets and joins of these subgraphs, the relation has an Induced concept lattice.

Incidence structure

These incidence structures have been generalized with block designs.
Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are on which lines for all points and lines.

Composition of relations

compositionrelation compositionrelative product
The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of A × B. In contrast to homogeneous relations, the composition of relations operation is only a partial function.
The opposite inclusion occurs for a difunctional relation.

Formal concept analysis

concept latticeconcept analysisconcept learning
Heterogeneous relations have been described through their induced concept lattices:
A data table that represents a heterogeneous relation between objects and attributes, tabulating pairs of the form "object g has attribute m", is considered as a basic data type.

Hyperbolic orthogonality

hyperbolic-orthogonal
3) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties.
Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a heterogeneous relation on sets of lines in the plane.

Logical matrix

(0,1)-matrixbinary matrixlogical matrices
Then the logical matrix for this relation is:
The outer product of P and Q results in an m × n rectangular relation:

Binary relation

relationrelationsidentity relation
In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B, where A and B are distinct sets.
To emphasize the fact X and Y are allowed to be different, a binary relation is also called a heterogeneous relation.

Outer product

bivector
A concept C ⊂ R satisfies two properties: (1) The logical matrix of C is the outer product of logical vectors
This type of matrix is used in the study of binary relations and is called a rectangular relation or a cross-vector.

Element (mathematics)

elementelementsset membership
The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation.
The relation "is an element of", also called set membership, is denoted by the symbol "\in".

Preorder

preordered setpreordered setsquasiorder
For a given relation R: X → Y, the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion \sqsubseteq forming a preorder.
Given a binary relation R, the complemented composition forms a preorder called the left residual, where denotes the converse relation of R, and denotes the complement relation of R, while \circ denotes relation composition.

Partial equivalence relation

In the context of homogeneous relations, a partial equivalence relation is difunctional.
Every partial equivalence relation is a difunctional relation, but the converse does not hold.

Jacques Riguet

Riguet, Jacques
In terms of the calculus of relations, in 1950 Jacques Riguet showed that such relations satisfy the inclusion
In 1950 he submitted "Sur les ensembles reguliers de relations binaires", and an article on difunctional relations, those with logical matrix in a block diagonal form.

Mathematics

mathematicalmathmathematician
In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B, where A and B are distinct sets.

Cartesian product

productCartesian squareCartesian power
In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B, where A and B are distinct sets.

Algebra of sets

set operationsalgebra of subsetsboolean
The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations.

Lattice (order)

latticelattice theorylattices
The inclusion R ⊆ S, meaning that aRb implies aSb, sets the scene in a lattice of relations.

Power set

powerset2all subsets
Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of A × B.

Partial function

total functionpartialtotal
In contrast to homogeneous relations, the composition of relations operation is only a partial function.

Category theory

categorycategoricalcategories
The necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations.

Category of sets

Setcategory of all setscategory of sets and functions
The necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations.

Morphism

morphismshom-setbimorphism
The necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations.

Category of relations

category of sets and relationsRelsets and relations
The objects of the category Rel are sets, and the relation-morphisms compose as required in a category.