Power series semigroup rings
John Dauns · Pacific Journal of Mathematics · 1970
A general method is given for constructing power series rings with exponents in a semigroup that need not be cancelϊative.Supports of power series need not be finite unions of inversely well ordered sets.1* Semigroups* The objective is to introduce the usual power series multiplication in subsets bigger than the semigroup ring.NOTATION 1.1.For a semigroup Γ and any ring R consider functions a: Γ-+R, or alternatively, a = X a(s)s with a(s) e R. Define the support of a-written as suppα:-to be the set supp = a -{s e Γ \ a(s) Φ 0} .Let P = P(Γ, R) = P(Γ) denote all a such that suppα is finite, i.e., the semigroup ring.1.2.If in addition Γ is a partially ordered set (notation: po-set), let W = W(Γ, R) = W(Γ) be the abelian group of all those a whose support supp a is the join of a finite number of inversely well ordered sets.