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LEE MONOIDS ARE NONFINITELY BASED WHILE THE SETS OF THEIR ISOTERMS ARE FINITELY BASED
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 28 March 2018, pp. 422-434
- Print publication:
- June 2018
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- Article
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We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids
$L_{\ell }^{1}$, obtained by adjoining an identity element to the semigroup generated by two idempotents 
$a$ and 
$b$ with the relation 
$0=abab\cdots \,$ (length 
$\ell$). We show that every monoid 
$M$ which generates a variety containing 
$L_{5}^{1}$ and is contained in the variety generated by 
$L_{\ell }^{1}$ for some 
$\ell \geq 5$ is nonfinitely based. We establish this result by analysing 
$\unicode[STIX]{x1D70F}$-terms for 
$M$, where 
$\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if 
$\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and 
$\ell \leq 5$, then the 
$\unicode[STIX]{x1D70F}$-terms (isoterms) for 
$L_{\ell }^{1}$ carry no information about the nonfinite basis property of 
$L_{\ell }^{1}$.
