Heterogeneous relation

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

A heterogeneous relation has been called a rectangular relation,^[1] suggesting that it does not have the square-symmetry of a relation on a set where A = B. This type of relation was termed heterorelativ in 1895 in an early exposition of the calculus of relations.^[2]

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

Examples

Oceans and continents

Ocean borders continent
	NA	SA	AF	EU	AS	OC	AA
Indian	0	0	1	0	1	1	1
Artic	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 = {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:

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) Given any graph of nodes and edges, a line graph can be formed with the set of nodes (A) and set of edges (B). The corresponding relation R ⊂ A × B holds (aRb) when node a is on the edge b. The contact aRb is expressed as incidence. A line graph is an example of an incidence structure. The logical matrix of an incidence structure is called an incidence matrix. The general study of incidence structures is called incidence geometry.

3) Another example arises in concept analysis where A is a set of "objects" and B is a set of "attributes", and where the meaning of relation R ⊂ A × B is that object a has attribute b when aRb. Jacques Riguet wrote of rectangular relations depending on two logical vectors

u=(u_{i})_{i=1}^{n}

and

v=(v_{i})_{i=1}^{m}

. The logical outer product of them generates an n × m logical matrix

a=(a_{ij}),\quad a_{ij}=u_{i}\land v_{j}.

Some authors write that matrix a corresponds to a rectangle in any relation that contains it.^[3] A concept in a relation R ⊂ A × B is a rectangle in R such that it is not contained in any larger rectangle. The Schein rank of a logical matrix M is the least number of outer products a for which the Boolean sum is M.^[4]

4) 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.

Particular relations

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

Difunctional

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

R\ R^{T}\ R\ \subseteq \ R.

^[5]

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.^[6]

In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management."^[7] Furthermore, bifunctional relations are fundamental in the study of bisimulations.^[8]

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.^[9]

Ferrers type

A strict order on a set is a homogeneous relation arising in order theory. For heterogeneous relations one requires that its logical matrix can be re-arranged to a staircase block form. An algebraic statement required for a Ferrers type relation R is

R{\bar {R}}^{T}R\subset R.

If any one of the relations $R,\ {\bar {R}},\ R^{T}$ is of Ferrer’s type, then all of them are. ^[10]

Relations of this type were named after N. M. Ferrers by Jacques Riguet^[11]

Contact

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 theory 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]

References

↑ Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.
↑ Ernst Schroder (1895) Algebra der Logik, Band III, page 12, via Internet Archive
↑ Ali Jaoua, Rehab Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in Relations and Kleene algebras in computer science, Lecture Notes in Computer Science 5827, Springer MR 2781235
↑ Ki Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0
↑ Jacques Riguet (1950) "Quelques proprietes des relations difonctionelles", Comptes Rendus 230: 1999–2000
↑ Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science & Business Media. p. 200. ISBN 978-3-211-82971-4.
↑ Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 in Relational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, and Gunther Schmidt, Springer Science & Business Media ISBN 978-3-211-82971-4
↑ Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences". Coalgebraic Methods in Computer Science. Lecture Notes in Computer Science. 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.
↑ Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.
↑ Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. p. 77. ISBN 978-3-642-77968-8.
↑ J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30
↑ Anne K. Steiner (1970) Review:Kontakt=Relationen from Mathematical Reviews

Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). "Chapter 3: Heterogeneous relations". Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. ISBN 978-3-642-77968-8.

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[1] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN 978-1-4020-6164-6.

[2] Ernst Schroder (1895) Algebra der Logik, Band III, page 12, via Internet Archive

[3] Ali Jaoua, Rehab Duwairi, Samir Elloumi, and Sadok Ben Yahia (2009) "Data mining, reasoning and incremental information retrieval through non enlargeable rectangular relation coverage", pages 199 to 210 in Relations and Kleene algebras in computer science, Lecture Notes in Computer Science 5827, Springer MR 2781235

[4] Ki Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0

[5] Jacques Riguet (1950) "Quelques proprietes des relations difonctionelles", Comptes Rendus 230: 1999–2000

[BrinkKahl1997-6] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science & Business Media. p. 200. ISBN 978-3-211-82971-4.

[7] Ali Jaoua, Nadin Belkhiter, Habib Ounalli, and Theodore Moukam (1997) "Databases", pages 197–210 in Relational Methods in Computer Science, edited by Chris Brink, Wolfram Kahl, and Gunther Schmidt, Springer Science & Business Media ISBN 978-3-211-82971-4

[8] Gumm, H. P.; Zarrad, M. (2014). "Coalgebraic Simulations and Congruences". Coalgebraic Methods in Computer Science. Lecture Notes in Computer Science. 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[Büchi1989-9] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions. Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[10] Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business Media. p. 77. ISBN 978-3-642-77968-8.

[11] J. Riguet (1951) "Les relations de Ferrers", Comptes Rendus 232: 1729,30

[12] Anne K. Steiner (1970) Review:Kontakt=Relationen from Mathematical Reviews