# 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.