Heterogeneous relation

It has been suggested that this article be merged into binary relation. (Discuss)

In mathematics, a heterogeneous relation is a subset of a Cartesian product A × B, where A and B are sets, possibly distinct.^[1]

A heterogeneous relation has been called a rectangular relation,^[2] suggesting that it does not have the square-symmetry of a relation on a set where A = B.

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 ${\displaystyle P\subseteq Q\equiv (P\cap {\bar {Q}}=\varnothing )\equiv (P\cap Q=P),}$ 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.

1) Let A = {Indian, Arctic, Atlantic, Pacific}, the oceans of the globe, and B = { NA, SA, AF, EU, AS, OC, AA }, the continents. Let aRb represent that ocean a borders continent b. Then the logical matrix for this relation is:

${\displaystyle R={\begin{pmatrix}0&0&1&0&1&1&1\\1&0&0&1&1&0&0\\1&1&1&1&0&0&1\\1&1&0&0&1&1&1\end{pmatrix}}.}$

The connectivity of the planet Earth can be viewed through R R^T and R^T R, the former being a 4 × 4 relation on A, which is the universal relation (A × A or a logical matrix of all ones). This universal relation reflects the fact that every ocean is separated from the others by at most one continent. On the other hand, R^T 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) An incidence structure (P, L, I) is a heterogeneous relation ${\displaystyle I\subseteq P\times L}$ where P is a set of points and L a set of lines and the relation I expresses incidence (point on line, line through point). The logical matrix of an incidence structure is called an incidence matrix. The general study of incidence structures is called incidence geometry.

3) Visualization of heterogeneous relations leans on graph theory: For relations on a set (homogeneous relations), a graph illustrates a corresponding relation. A bipartite graph is used to illustrate a heterogeneous relation. Just as the clique is integral to relations on a set, so bicliques are used to describe heterogeneous relations; indeed, the rectangles in a heterogeneous relation are illustrated through bicliques in the corresponding bipartite graph.

Heterogeneous relations have been described through their induced concept lattices: A concept C ⊂ R satisfies two properties: (1) The logical matrix of C is the outer product of logical vectors

${\displaystyle C_{ij}\ =\ u_{i}v_{j},\quad u,v}$ logical vectors. (2) C is maximal, not contained in any other outer product. Thus C is described as a non-enlargeable rectangle.

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 ${\displaystyle \sqsubseteq }$ forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".^[3] The decomposition is

${\displaystyle R\ =\ f\ E\ g^{T},}$ where f and g are functions, called mappings or left-total, univalent relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order E that belongs to the minimal decomposition (f, g, E) of the relation R."

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.^[4] Structural analysis of relations with concepts provides an approach for data mining.^[5]

• Proposition: If R is a total relation and R^T is its transpose, then ${\displaystyle I\subseteq R^{T}R}$ where I is the m × m identity relation.
• Proposition: If R is a surjective relation, then ${\displaystyle I\subseteq RR^{T}}$ where I is the n × n identity relation.

Among the homogeneous relations on a set, the equivalence relations are distinguished for their ability to partition the set. With heterogeneous relations the idea is to partition objects by distinguishing attributes. One way this can be done is with an intervening set Z = {x, y, z, ...} of indicators. The partitioning relation R = F G^T is a composition of relations using univalent relations F ⊆ A × Z and G ⊆ B × Z.

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

${\displaystyle R\ R^{T}\ R\ \subseteq \ R.}$^[6]

He named these relations difunctional since the composition F G^T involves univalent relations, commonly called functions.

Using the notation {y: xRy} = 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.^[7]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."^[8] Furthermore, bifunctional relations are fundamental in the study of bisimulations.^[9]

In the context of homogeneous relations, a partial equivalence relation is bifunctional.

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 (asymmetric) main diagonal.^[10]

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.^[11]

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

${\displaystyle R{\bar {R}}^{T}R\subseteq R.}$

If any one of the relations ${\displaystyle R,\ {\bar {R}},\ R^{T}}$ is of Ferrers type, then all of them are. ^[1]

Suppose B is the power set of A, the set of all subsets of A. Then a contact relation g satisfies three properties: (1) ∀ x in A, Y = {x} implies x g Y. (2) Y ⊆ Z and x g Y implies x g Z. (3) ∀ 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 (1970).^[12]

In terms of the calculus of relations, sufficient conditions for a contact relation include

${\displaystyle C^{T}{\bar {C}}\ \subseteq \ \ni {\bar {C}}\ \ \equiv \ C\ {\overline {\ni {\bar {C}}}}\ \subseteq \ C,}$ where ${\displaystyle \ni }$ is the converse of set membership (∈).^[13]: 280

Every relation R generates a preorder R\R which is the left residual. In terms of converse and complements, ${\displaystyle R\backslash R\ \equiv \ {\overline {R^{T}{\bar {R}}}}.}$ Forming the diagonal of ${\displaystyle R^{T}{\bar {R}}}$, the corresponding row of R^T and column of ${\displaystyle {\bar {R}}}$ will be of opposite logical values, so the diagonal is all zeros. Then

${\displaystyle R^{T}{\bar {R}}\subseteq {\bar {I}}\ \implies \ I\subseteq {\overline {R^{T}{\bar {R}}}}\ =\ R\backslash R,}$ so that R\R is a reflexive relation.

To show transitivity, one requires that (R\R)(R\R) ⊂ R\R. Recall that X = R\R is the largest relation such that R X ⊂ R. Then

${\displaystyle R(R\backslash R)\subseteq R}$

${\displaystyle R(R\backslash R)(R\backslash R)\subseteq R}$ (repeat)

${\displaystyle \equiv R^{T}{\bar {R}}\subseteq {\overline {(R\backslash R)(R\backslash R)}}}$ (Schröder's rule)

${\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq {\overline {R^{T}{\bar {R}}}}}$ (complementation)

${\displaystyle \equiv (R\backslash R)(R\backslash R)\subseteq R\backslash R.}$ (definition)

The inclusion relation Ω on the power set of U can be obtained in this way from the membership relation ∈ on subsets of U:

${\displaystyle \Omega \ =\ {\overline {\ni {\bar {\in }}}}\ =\ \in \backslash \in .}$^[13]: 283

Given a relation R, a sub-relation called its fringe is defined as

${\displaystyle \operatorname {fringe} (R)=R\cap {\overline {R{\bar {R}}^{T}R}}.}$

When R is a partial identity relation, difunctional, or a block diagonal relation, then fringe(R) = R. Otherwise the fringe operator selects a boundary sub-relation described in terms of its logical matrix: fringe(R) is the side diagonal if R is an upper right triangular linear order or strict order. Fringe(R) is the block fringe if R is irreflexive (${\displaystyle R\subseteq {\bar {I}}}$) or upper right block triangular. Fringe(R) is a sequence of boundary rectangles when R is of Ferrers type.

On the other hand, Fringe(R) = ∅ when R is a dense, linear, strict order.^[13]

• Category of relations
• Allegory (category theory)

• Schmidt, Gunther; Ströhlein, Thomas (2012). "Chapter 3: Heterogeneous relations". Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.
• Ernst Schröder (1895) Algebra der Logik, Band III, via Internet Archive</ref>