Heterogeneous 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. The prefix hetero is from the Greek ἕτερος.
A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-symmetry of a homogeneous relation on a set where A = B. Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between different sets."
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. The inclusion R ⊆ S, meaning that aRb implies aSb, sets the scene in a lattice of relations. But since the inclusion symbol is superfluous. 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 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. The objects of the category Rel are sets, and the relation-morphisms compose as required in a category.
Examples
| NA | SA | AF | EU | AS | OC | AA | |
| Indian | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| Arctic | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| Atlantic | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
| Pacific | 1 | 1 | 0 | 0 | 1 | 1 | 1 |
1) Let A =, the oceans of the globe, and B =, the continents. Let aRb represent that ocean a borders continent b. Then the logical matrix for this relation is:
The connectivity of the planet Earth can be viewed through R RT and RT R, the former being a 4 × 4 relation on A, which is the universal relation. This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, RT R is a relation on B × B which fails to be universal because at least two oceans must be traversed to voyage from Europe to Oceania.
2) Visualization of relations leans on graph theory: For relations on a set, a directed graph illustrates a relation and a graph a symmetric relation. For heterogeneous relations a hypergraph has edges possibly with more than two nodes, and can be illustrated by a bipartite graph.
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.
3) Hyperbolic orthogonality: Time and space are different categories, and temporal properties are separate from spatial properties. The idea of simultaneous events is simple in absolute time and space since each time t determines a simultaneous hyperplane in that cosmology. Herman Minkowski changed that when he articulated the notion of relative simultaneity, which exists when spatial events are "normal" to a time characterized by a velocity. He used an indefinite inner product, and specified that a time vector is normal to a space vector when that product is zero. The indefinite inner product in a composition algebra is given by
As a relation between some temporal events and some spatial events, hyperbolic orthogonality is a heterogeneous relation.
4) A geometric configuration can be considered a relation between its points and its lines. The relation is expressed as incidence. Finite and infinite projective and affine planes are included. Jakob Steiner pioneered the cataloguing of configurations with the Steiner systems S which have an n-element set S and a set of k-element subsets called blocks, such that a subset with t elements lies in just one block. These incidence structures have been generalized with block designs. The incidence matrix used in these geometrical contexts corresponds to the logical matrix used generally with binary relations.
Induced concept lattice
Heterogeneous relations have been described through their induced concept lattices:A concept C ⊂ R satisfies two properties: The logical matrix of C is the outer product of logical vectors
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 forming a preorder.
The MacNeille completion theorem is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is
Particular cases are considered below: E total order corresponds to Ferrers type, and E identity corresponds to difunctional, a generalization of equivalence relation on a set.
Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining.
Particular relations
- Proposition: If R is a total relation and RT is its transpose, then where I is the m × m identity relation.
- Proposition: If R is a surjective relation, then where I is the n × n identity relation.
Difunctional
The logical matrix of such a relation R can be re-arranged as a block matrix with blocks of ones along the diagonal.
In terms of the calculus of relations, in 1950 Jacques Riguet showed that such relations satisfy the inclusion
He named these relations difunctional since the composition F GT involves univalent relations, commonly called functions.
Using the notation = xR, a difunctional relation can also be characterized as a relation R such that wherever x1R and x2R have a non-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.
In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management." Furthermore, difunctional relations are fundamental in the study of bisimulations.
In the context of homogeneous relations, a partial equivalence relation is difunctional.
In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block diagonal matrix with rectangular blocks of true on the main diagonal.
Ferrers type
A strict order on a set is a homogeneous relation arising in order theory.In 1951 Jacques Riguet adopted the ordering of a partition of an integer, called a Ferrers diagram, to extend ordering to heterogeneous relations.
The corresponding logical matrix of a heterogeneous relation has rows which finish with a non-increasing sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.
An algebraic statement required for a Ferrers type relation R is
If any one of the relations is of Ferrers type, then all of them are.
Contact
Suppose B is the power set of A, the set of all subsets of A. Then a contact relation g satisfies three properties: ∀ x in A, Y = implies x g Y. Y ⊆ Z and x g Y implies x g Z. ∀ y in Y, y g Z and x g Y implies x g Z. The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in his book Kontakt-Relationen.In terms of the calculus of relations, sufficient conditions for a contact relation include
Preorder R\R
Every relation R generates a preorder R\R which is the left residual. In terms of converse and complements, Forming the diagonal of, the corresponding row of RT and column of will be of opposite logical values, so the diagonal is all zeros. ThenTo show transitivity, one requires that ⊂ R\R. Recall that X = R\R is the largest relation such that R X ⊂ R. Then
The inclusion relation Ω on the power set of U can be obtained in this way from the membership relation ∈ on subsets of U:
Fringe of a relation
Given a relation R, a sub-relation called its fringe is defined asWhen R is a partial identity relation, difunctional, or a block diagonal relation, then fringe = R. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe is the side diagonal if R is an upper right triangular linear order or strict order. Fringe is the block fringe if R is irreflexive or upper right block triangular. Fringe is a sequence of boundary rectangles when R is of Ferrers type.
On the other hand, Fringe = ∅ when R is a dense, linear, strict order.