Important examples of such tensors include symmetric and antisymmetric tensors. This map exhibits as a totally antisymmetric covariant tensor field of rank . This compatibility condition is neither transitive, symmetric nor antisymmetric. A related concept is that of the antisymmetric tensor or alternating form. Let formula 86 be an antisymmetric tensor of rank "r". Particles which exhibit antisymmetric states are called fermions. The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. In the theory of quantum mechanics fermions are described by antisymmetric states. Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. A preorder is a relation that is reflexive and transitive, but not necessarily antisymmetric. Let formula 25 be odd polynomials in some antisymmetric variables formula 26. An "partial order", also called "order", is a relation that is reflexive, antisymmetric, and transitive. The divisibility relation on the natural numbers is an important example of an antisymmetric relation. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric. The symmetrical and antisymmetric zero-order modes deserve special attention.