  We introduce a new variety of set-theoretic non-associative algebras,
P{\l}onka bi-magmas, to describe and classify all solutions of the
set-theoretic Yang-Baxter (YB) equation of Baaj-Long-Skandalis (BLS) type. We
also study new classes of YB-solutions (bi-connected, simple), and classify the
BLS-solutions that fit into those classes. There are only countably many
isomorphism classes of simple BLS-solutions, for instance, and they are all
describable in terms of the odometer transformations familiar from ergodic
theory.
  Placing Drinfel'd's problem of classifying the set-theoretic solutions of the
Yang-Baxter equation in a universal-algebra context with a combinatorial
flavor, we also prove the existence of adjunctions between various categories
of solutions.
