Partial equivalence relation

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In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) on a set is a binary relation that is symmetric and transitive. In other words, it holds for all that:

  1. if , then (symmetry)
  2. if and , then (transitivity)

If is also reflexive, then is an equivalence relation.

Properties and applications[edit]

If a relation on a set is a PER then is an equivalence relation on the subset .[note 1] However, given a set , a relation on whose restriction to a given subset is an equivalence on need not be a PER on ; for instance, considering the set , the relation over characterised by the set is an equivalence relation on but not a PER on since it is neither symmetric[note 2] nor transitive[note 3] on .

Every partial equivalence relation is a difunctional relation, but the converse does not hold.

Each partial equivalence relation is a right Euclidean relation. The contrary does not hold: for example, xRy on natural numbers, defined by 0 ≤ xy+1 ≤ 2, is right Euclidean, but neither symmetric (since e.g. 2R1, but not 1R2) nor transitive (since e.g. 2R1 and 1R0, but not 2R0). Similarly, each partial equivalence relation is a left Euclidean relation, but not vice versa. Each partial equivalence relation is quasi-reflexive,[1] as a consequence of being Euclidean.

In non-set-theory settings[edit]

In type theory, constructive mathematics and their applications to computer science, constructing analogues of subsets is often problematic[2]—in these contexts PERs are therefore more commonly used, particularly to define setoids, sometimes called partial setoids. Forming a partial setoid from a type and a PER is analogous to forming subsets and quotients in classical set-theoretic mathematics.

The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive.[3]

Examples[edit]

A simple example of a PER that is not an equivalence relation is the empty relation , if is not empty.

Kernels of partial functions[edit]

If is a partial function on a set , then the relation defined by

if is defined at , is defined at , and

is a partial equivalence relation, since it is clearly symmetric and transitive.

If is undefined on some elements, then is not an equivalence relation. It is not reflexive since if is not defined then — in fact, for such an there is no such that . It follows immediately that the largest subset of on which is an equivalence relation is precisely the subset on which is defined.

Functions respecting equivalence relations[edit]

Let X and Y be sets equipped with equivalence relations (or PERs) . For , define to mean:

then means that f induces a well-defined function of the quotients . Thus, the PER captures both the idea of definedness on the quotients and of two functions inducing the same function on the quotient.

Equality of IEEE floating point values[edit]

The IEEE 754:2008 floating point standard defines an "EQ" relation for floating point values. This predicate is symmetrical and transitive, but is not reflexive because of the presence of NaN values that are not EQ to themselves.

Notes[edit]

  1. ^ By construction, is reflexive on and therefore an equivalence relation on . Actually, can hold only on elements of : if , then by symmetry, so and by transitivity, that is, .
  2. ^ , but not
  3. ^ and , but not

References[edit]

  1. ^ Encyclopaedia Britannica (EB); although EB's and Wikipedia's notions of quasi-reflexivity differ in general, they coincide for symmetric relations.
  2. ^ Salveson, A.; Smith, J.M. (1988). "The strength of the subset type in Martin-Lof's type theory". [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science. pp. 384–391. doi:10.1109/LICS.1988.5135. ISBN 0-8186-0853-6. S2CID 15822016.
  3. ^ J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini; Paulo Agliano (eds.). Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.