On the Imbedding of a Finite Commutative Semigroup of Idempotents in a Uniquely Factorable Semigroup.

A semigroup is a set of elements to which is related an operation usually called multiplication and an equivalence relation, such that the set is closed and associative relative to the operation. We shall discuss, briefly, finite semigroups which are uniquely factorable in the same sense as the multiplicative semigroup of all nonzero integers. Clifford' defined an arithmetic in such a way as to include our uniquely factorable semigroups as well as similar infinite commutative semigroups, not necessarily cancellable, and gave necessary and sufficient conditions for a commutative sernigroup to be imbedded in an arithmetic. He used ideals to accomplish this imbedding. We shall accomplish a constructive imbedding of any finite commutative semigroup of idempotents alone in a uniquely factorable semigroup, by the use of correspondences, or generalized substitutions. This method of imbedding is the chief novelty herein. Let a finite commutative semigroup S contain an identity, and U be the set of units, or divisors of the identity. Then U is a group, and S can be divided exhaustively into disjoint cosets2 relative to U. This division into cosets is unique, and the cosets form a semigroup, designated by S/U, whose identity is U, such that S/U is a homomorphic image of S. We shall use = and ; as the equivalence relations of S/U and S, respectively. If s e S, the elements of the coset sU are called associated elements or associates. If S is a finite semigroup, a e S, and k and s are the least positive integers such that k < s and ak o a', then s 1, k, and s-k are called, respectively, the order, index, and period of a. The semigroup S is called a uniquely factorable semigroup, abbreviated "UFS," provided that S is finite and commutative and contains at least two elements, one of which is a unit, and satisfies: