By Professor Richard Hubert Bruck (auth.)

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Fortheröle of semigroups in analysis see RILLE, Functional Analysis and Semigroups. Aside from this reference, the Iiterature on the application of semigroups to analysis or topology has been omitted from the bibliography. Likewise, papers applying semigroup theory to arithmetic (unique factorization) and lattice theory have largely been omitted. The homomorphism theory of semigroups has been studied rather thoroughly without many interesting results. If 0 is a homomorphism of a semigroup S into a groupoid, the image S 0 = S' is also a semigroup.

Hence a group G has a unique identity element. That is: a group is an associative loop. The identity of G will be denoted by 1. If a E G, elements a', a" of G are uniquely defined by aa'= 1, a"a = 1. Then a"= a"1 = a"(aa') = (a" a) a' = 1 a' = a'. The element a" = a' is called the inverse of a and denoted by a- 1. As is easily verified, (a- 1)- 1= a and (ab)- 1 = b-1a-t. 1. Let G be a semigroup with a right identity elementesuch that to each a E G there corresponds at least one element a' E G satisfying aa' = e.

When (i), (ii) hold we may imbed M in a ring F as follows: Let F be the additive free abelian group with the elements of M as free basis. 1) for all a, b, c in M. 1) holds for non-negative integers n(a, b, c), not all zero, we make M into a multigroupoid by defining ab to be the subset of M consisting of all c in M for which n (a, b, c) > 0. For example, let R be the ring defined as above from a quasigroup G. Certain subrings of R give rise quite naturally to multigroupoids with multiplicity functions.