is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups. Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937. Semigroup methods in partial differential equations Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval and times : Let be the Lp space of square-integrable real-valued functions with domain the interval and let A be the second-derivative operator with domain where H2 is a Sobolev space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X: On an heuristic level, the solution to this problem "ought" to be . However, for a rigorous treatment, a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X to itself, taking the initial state u0 at time to the state at time t. The operator A is said to be the infinitesimal generator of the semigroup. History The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groups or rings. A number of sources attribute the first use of the term (in French) to J.-A. de Séguier in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order. Anton Sushkevich obtained the first non-trivial results about semigroups. His 1928 paper "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" ("On finite groups without the rule of unique invertibility") determined the structure of finite simple semigroups and showed that the minimal ideal (or Green's relations J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by David Rees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. Clifford and Gordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In