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My question again refers to the following article: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...

$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure: $R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation": $$R_X (x, y) \cdot R_Y (x +...

This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: ...