In modern algebra, an antiisomorphism (or anti-isomorphism) between structured sets

Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

 A and B is an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 from A to the opposite

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

 of B (or equivalently from the opposite of A to B). If there exists an antiisomorphism between two structures, they are antiisomorphic.

Intuitively, to say that two algebraic structures are antiisomorphic is to say that they are basically opposites of one another.

An example may clarify the idea. Let A be the binary relation (or graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

) consisting of elements {1,2,3} and binary relation defined as follows:

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Let B be the binary relation set consisting of elements {a,b,c} and binary relation defined as follows:

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Note that the opposite of B (called B^op) is the same set of elements with the opposite binary relation :

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If we replace a, b, and c with 1, 2, and 3 respectively, we will see that each rule in B^op is the same as some rule in A. That is, we can define an isomorphism from A to B^op by



This is an antiisomorphism between A and B.