Maths - Setoid

Setoid

In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~.

Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set). - From Wiki

One way to define a setoid is define a tuple consisting of:

For an Idris implementation of setoids see this github site

Relationship to Groupoid

  Group Definition Equivalence Definition
Identity a = a•Id = Id •a

reflexive

a~a

Reverse a•inv(a) = inv(a)•a = Id

symmetric

if a~b then b~a

Associatively a•(b•c) = (a•b)•c

transitive

if a~b and b~c then a~c

Groupoid is explained on this page.

A central idea in the definition of setoid is not to insist upon equality between proof objects.

Extensionality of functions

if f(x) = g(x) for all x

then f = g

Example - consider the two functions f and g mapping from and to natural numbers, defined as follows:

These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same.

Functions between Setoids

Extensional Functions between Setoids

A function between setoids is a function between the underlying types (sets) together with a proof of extensionality:

A function |f| : |X|->|Y| is extensional, with respect to the equalities of X and Y,if there is a term of type:

f ~X=>Y g: Π|x|,|x'|:|X| x ~X x' -> |f|(x) ~Y |f|(y)

The bars, as in |x|, indicate x is an element of the underlying type. This distinction will usually be dropped if extensionality applies.

x

 

 


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