Bijective 1-cocycles, braces, and non-commutative prime factorization

Abstract

Summary: The structure group of an involutive set-theoretic solution to the Yang-Baxter equation is a generalized radical ring called a brace . The concept of brace is extended to that of a quasiring where the adjoint group is just a monoid. It is proved that a special class of lattice-ordered quasirings characterizes the divisor group A of a smooth non-commutative curve X . The multiplicative monoid A ◦ of A is related to the additive group by a bijective 1-cocycle. Extending previous results on non-commutative arithmetic, the elements of A are represented as a class Φ ( X ) of self-maps of a universal cover of X . For affine subsets U of X , the regular functions on U form a hereditary order such that the monoid of fractional ideals embeds into A ◦ as the class of monotone functions in Φ ( U ) . The unit group of A is identified with the annular symmetric group , which occurred in connection with quasi-Garside groups of Euclidean type. The main part of the paper is self-contained and provides a quick approach to non-commutative prime factorization and its relationship to braces.