antisimetri

Antisimetri, or antisymmetry, is a mathematical concept describing a relation that cannot hold in both directions between distinct elements. Formally, a relation R on a set A is antisymmetric if for all a, b in A, if aRb and bRa, then a = b. Equivalently, there are no distinct elements a and b with both aRb and bRa. Antisymmetry is a key property of partial orders; all partial orders are antisymmetric, and a total (linear) order is antisymmetric as well. Examples include the usual numerical order ≤ on real numbers: if a ≤ b and b ≤ a, then a = b. The subset relation ⊆ on sets is antisymmetric: if A ⊆ B and B ⊆ A, then A = B. The divisibility relation on positive integers is antisymmetric: if a divides b and b divides a, then a = b.

Antisymmetry is distinct from asymmetry. A relation is asymmetric if aRb implies not bRa; asymmetric relations are automatically antisymmetric but not vice versa.

In linear algebra, the term antisymmetric (often called skew-symmetric) is used for bilinear forms B with B(v, w) = -B(w, v) for all vectors v and w. The matrix representation A of a skew-symmetric form satisfies A^T = -A; over fields of characteristic not 2, this implies all diagonal entries are zero. Skew-symmetric forms arise in geometry and physics, such as in the study of symplectic structures.

Terminology can vary: some authors reserve antisymmetric for the relational property and use skew-symmetric for bilinear forms; others use antisymmetric to cover both notions.