Some other techniques for studying semigroups, like Green's relations, do not resemble anything in group theory. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov processes. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There are also interesting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kind are bands and their commutative subclass—semilattices, which are also ordered algebraic structures. Definition A semigroup is a set together with a binary operation "" (that is, a function ) that satisfies the associative property: For all , the equation holds. More succinctly, a semigroup is an associative magma. Examples of semigroups Empty semigroup: the empty set forms a semigroup with the empty function as the binary operation. Semigroup with one element: there is essentially only one (specifically, only one up to isomorphism), the singleton {a} with operation . Semigroup with two elements: there are five which are essentially different. The "flip-flop" monoid: a semigroup with three elements representing the three operations on a switch - set, reset, and do nothing. The set of positive integers with addition. (With 0 included, this becomes a monoid.) The set of integers with minimum or maximum. (With positive/negative infinity included, this becomes a monoid.) Square nonnegative matrices of a given size with matrix multiplication. Any ideal of a ring with the multiplication of the ring. The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation — the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. A probability distribution F together with all convolution powers of F, with convolution as the operation. This is called a convolution semigroup. Transformation semigroups and monoids. The