Monomial algebras defined by Lyndon words

Assume that $X=x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_g$ is a finite alphabet and $K$ is a field. We study monomial algebras $A=K<X>\mathrm{/}(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincar'{e}-Birkhoff-Witt type basis of $A$ in terms of its \emph{Lyndon atoms} $N$, but, in general, $N$ may be infinite. We prove that if $A$ has polynomial growth of degree $d$ then $A$ has global dimension $d$ and is standard finitely presented, with $d-1 \leq |W| \leq d(d-1)/2$\mathrm{.}Furthermore,$A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\mathfrak{N} = {l_1^{\alpha_{1}}l_2^{\alpha_{2}}… l_d^{\alpha_{d}} \mid \alpha_{i} \geq 0, 1 \leq i \leq d}$,where$N = {l_1, …,l_d}$. We give an extremal class of monomial algebras, the Fibonacci-Lyndon algebras, $F_n$, with global dimension $n$ and polynomial growth, and show that the algebra $F_6$ of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular algebra.