Gigel Militaru
University of Bucharest, Bucharest, Romania

We prove that the category of solutions of the set-theoretic quantum Yang-Baxter equation of Frobenius type is equivalent to the category of pointed Kimura semigroups.

As applications, all nondegenerate, idempotent, bijective, involutive, finite order or unitary solutions are classified. For instance, if $|X| = n$, then the number of isomorphism classes of all such solutions on the set $X$ that are (a) left non-degenerate, (b) bijective or (c) unitary is: (a) the Davis number $d(n)$, (b) $\sum_{m|n} \, p(m)$, where $p(m)$ is the partition number, or (c) $\tau(n) + \sum_{d|n}\left\lfloor \frac d2\right\rfloor$, where $\tau(n)$ is the number of divisors of $n$. Several others applications are given. Joint work with Ana Agore and Alexandru Chirvasitu.