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duality
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A diagram illustrating projective geometry duality, showing a point labeled P and a line labeled RS intersecting at an oval shape, with arrows indicating the association between point P and line RS.
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duality

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duality, in mathematics, principle whereby one true statement can be obtained from another by merely interchanging two words. It is a property belonging to the branch of algebra known as lattice theory, which is involved with the concepts of order and structure common to different mathematical systems. A mathematical structure is called a lattice if it can be ordered in a specified way (see order). Projective geometry, set theory, and symbolic logic are examples of systems with underlying lattice structures, and therefore also have principles of duality.

Projective geometry has a lattice structure that can be seen by ordering the points, lines, and planes by the inclusion relation. In the projective geometry of the plane, the words “point” and “line” can be interchanged, giving for example the dual statements: “Two points determine a line” and “Two lines determine a point.” This last statement, sometimes false in Euclidean geometry, is always true in projective geometry because the axioms do not allow for parallel lines. Sometimes the language of a statement must be modified in order that the corresponding dual statement be clear; the dual of the statement “Two lines intersect in a point” is vague, while the dual of “Two lines determine a point” is clear. Even the statement “Two points intersect in a line,” however, can be understood if a point is considered as a set (or “pencil”) containing all the lines on which it lies, a concept itself dual to the idea of a line being considered as the set of all points that lie on it.

There is a corresponding duality in three-dimensional projective geometry between points and planes. Here, the line is its own dual, because it is determined by either two points or two planes.

In set theory, the relations “contained in” and “contains” can be interchanged, with the union becoming the intersection and vice-versa. In this case, the original structure remains unchanged, so it is called self-dual.

In symbolic logic there is a similar self-duality if “implied” and “is implied by” are interchanged, along with the logical connectives “and” and “or.”

Duality, a pervasive property of algebraic structures, holds that two operations or concepts are interchangeable, all results holding in one formulation also holding in the other, the dual formulation.