Heterogeneous relation

Oceans and continents

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) Visualization of relations leans on graph theory: For relations on a set (homogeneous relations), 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.

The various t axes represent time for observers in motion, the corresponding x axes are their lines of simultaneity

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

${\displaystyle <x,z>\ =\ x{\bar {z}}+{\bar {x}}z}$  where the overbar denotes conjugation.

As a relation between some temporal events and some spatial events, hyperbolic orthogonality (as found in split-complex numbers) is a heterogeneous relation.[4]

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(t,k,n) 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.

An incidence structure is a triple D = (V, B, I) where V and B are any two disjoint sets and I is a binary relation between V and B, i.e. I ⊆ V × B. The elements of V will be called points, those of B blocks and those of I flags.[5]

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".[6] 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.[7] Structural analysis of relations with concepts provides an approach for data mining.[8]

• 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.

Every relation R generates a preorder R\R which is the left residual.[17] 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:

${\displaystyle \Omega \ =\ {\overline {\ni {\bar {\in }}}}\ =\ \in \backslash \in .}$[16]^: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.[16]

Given two sets A and B, the set of binary relations between them ${\displaystyle {\mathcal {B}}(A,B)}$ can be equipped with a ternary operation ${\displaystyle [a,\ b,\ c]\ =\ ab^{T}c}$ where b^T denotes the converse relation of b. In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps.[18][19] The contrast of heterogeneous and homogeneous relations is highlighted by these definitions:

There is a pleasant symmetry in Wagner's work between heaps, semiheaps, and generalised heaps on the one hand, and groups, semigroups, and generalised groups on the other. Essentially, the various types of semiheaps appear whenever we consider binary relations (and partial one-one mappings) between different sets A and B, while the various types of semigroups appear in the case where A = B.[20]

• 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