Dynamical extensions of quasi-linear left cycle sets and the Yang–Baxter equation

Abstract

In this paper, we give a characterization of the quasi-linear left cycle sets [Formula: see text] with [Formula: see text] via unitary metahomomorphisms and a complete description of those with [Formula: see text] improving the results obtained in [F. Catino and M. M. Miccoli, Construction of quasi-linear left cycle sets, J. Algebra Appl. 14(1) (2015), Article ID:1550001, 1–7]. Moreover, we develop a theory of dynamical extensions of quasi-linear left cycle sets to provide new set-theoretic solutions of the Yang–Baxter equation that are non-degenerate, involutive and multipermutational.