Cale Monoids, Cale Domains, and Cale Varieties
In an integral domain or monoid, two diﬀerent factorizations of an element into atoms often turn out to be essentially the same when lifted to powers. In this chapter, we analyze this fascinating phenomenon, which surprisingly occurs for a great many algebraic structures. These structures range from algebraic number rings over Diophantine monoids, to varieties and their deﬁning semigroup rings. Analytically we capture this phenomenon by the concept of a Cale representation, which is a kind of unique representation with respect to powers. Section 1 of this chapter presents the exact deﬁnitions concerning Cale concepts, as well as an array of rather diﬀerent examples which illustrate the lifting of nonunique factorization to unique Cale representation. In Section 2, we consider Cale representation with respect to extraction analysis, which is one of the key tools used in the arguments throughout this chapter. Section 3 considers some useful special elements suggested by the study of the Cale property and places these elements in context with respect to the class of Krull monoids. In Section 4 we present for monoids, as well as for domains, general conditions for the Cale representation to exist. We take mainly from a joint work of the current authors with Franz Halter-Koch . In Section 5 we present particular results on Cale representation in algebraic orders, Diophantine monoids and semigroup rings. Much of this work is drawn from the papers [4, 18, 21, 22]. In Section 6 we investigate aﬃne toric varieties deﬁned via semigroup rings for Cale monoids. The main result (Theorem 6.5) gives a description of those varieties by only a few binomials stemming from Cale representation.