These congruence classes are precisely the ones which have modular multiplicative inverses. When this notation is used, inverse functions could be confused with multiplicative inverses. As was mentioned above, commutative rings satisfy all axioms of fields, except for multiplicative inverses. Cayley defined matrix multiplication and matrix inverses. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. These considerations are particularly important for defining the inverses of trigonometric functions. For example, the additive and multiplicative inverses and are uniquely determined by . These axioms are closure, associativity, and the inclusion of an identity element and inverses. Negative iterates correspond to function inverses and their compositions. This article describes generalized inverses of a matrix formula 1. Thus the elements of the subspace do not have inverses. Several techniques can be used to explicitly construct reversible cellular automata with known inverses. This iteration can also be generalised to a wider sort of inverses, e.g. matrix inverses. Analytic properties of functions pass to their inverses.