The classification of Artin-Schelter regular algebras and finding new classes of regular algebras are fundamental problems

in noncommutative algebraic geometry. In this talk we consider classes *C(X, W)* of associative graded algebras *A* over a

field *K*, generated by a finite set *X* and with a fixed obstructions set *W*, where *W* is a finite antichain of Lyndon words

in *X*. The main question is: when such a class contains an AS regular algebra? Given the global dimension *d* of *A*, we

that the order of W satisfies* *the inequality d-1 ≤ |W| ≤ d(d-1)/2. We prove that W is unique, and *C(X, W) *contains regular

algebras, whenever |W| attains the lower, or the upper bounds. If the order of *W* is *d(d-1)/2,* *C(X, W)* contains numerous

non-isomorphic PBW regular algebras, each of which defines a set-theoretic solution of the Yang-Baxter equation.

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