1970, a new periodical called Semigroup Forum (currently edited by Springer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory. The representation theory of semigroups was developed in 1963 by Boris Schein using binary relations on a set A and composition of relations for the semigroup product. At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations on A. In 1997 Schein and Ralph McKenzie proved that every semigroup is isomorphic to a transitive semigroup of binary relations. In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like inverse semigroups, as well as monographs focusing on applications in algebraic automata theory, particularly for finite automata, and also in functional analysis. Generalizations If the associativity axiom of a semigroup is dropped, the result is a magma, which is nothing more than a set M equipped with a binary operation that is closed . Generalizing in a different direction, an n-ary semigroup (also n-semigroup, polyadic semigroup or multiary semigroup) is a generalization of a semigroup to a set G with a n-ary operation instead of a binary operation. The associative law is generalized as follows: ternary associativity is , i.e. the string abcde with any three adjacent elements bracketed. N-ary associativity is a string of length with any n adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an n-ary group. A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities. Infinitary generalizations of commutative semigroups have sometimes been considered by various authors. See also Absorbing element Biordered set Empty semigroup Generalized inverse Identity element Light's associativity test Quantum dynamical semigroup Semigroup ring Weak inverse Notes Citations References General references Specific references Semigroup theory Algebraic structures