We establish a new sufficient condition under which a monoid is non-finitelybased and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoiningan identity element to the semigroup generated by two idempotents $a$ and $b$subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and iscontained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ isnon-finitely based.
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