科研之友
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摘要
Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k is an element of N, let U-k(H) denote the set of all l is an element of N with the property that there are atoms u(1), ... , u(k), v(1), ... , v(l) such that u(1) ..... u(k) = v(1) .... v(l) (thus, U-k(H) is the union of all sets of lengths containing k). The Structure Theorem for Unions states that, for all sufficiently large k, the sets U-k(H) are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.
- 出版日期2017-12
- 单位中国地质大学（北京）
