• An example comes from reversing the direction of inequalities in a
partial order. So if
X is a
set and ≤ a partial order relation, we can define a new partial order relation ≤op by ::
x ≤op
y if and only if
y ≤
x. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore,
duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor,
infimum/
supremum,
down-set/
up-set,
ideal/
filter etc. This order theoretic duality is in turn a special case of the construction of opposite categories as every ordered set can be
understood as a category. • Given a
semigroup (
S, ·), one usually defines the opposite semigroup as (
S, ·)op = (
S, *) where
x*
y ≔
y·
x for all
x,
y in
S. So also for semigroups there is a strong duality principle. Clearly, the same construction works for groups, as well, and is known in
ring theory, too, where it is applied to the multiplicative semigroup of the ring to give the opposite ring. Again this process can be described by completing a semigroup to a monoid, taking the
corresponding opposite category, and then possibly removing the unit from that monoid. • The category of
Boolean algebras and Boolean
homomorphisms is
equivalent to the opposite of the category of
Stone spaces and
continuous functions. • The category of
affine schemes is
equivalent to the opposite of the category of
commutative rings. • The
Pontryagin duality restricts to an equivalence between the category of
compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups. • By the Gelfand–Naimark theorem, the category of localizable
measurable spaces (with
measurable maps) is equivalent to the category of commutative
Von Neumann algebras (with
normal unital homomorphisms of
*-algebras). ==Properties==