Bijective 1-cocycles, braces, and non-commutative prime factorization
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^\circ $ 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 $\Phi (X)$ 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^\circ $ as the class of monotone functions in $\Phi (U)$. 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.