Lee monoids are non-finitely based while the sets of their isoterms are finitely based

2016/10/30 by Sapir, Olga
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.09721

Abstract

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids L_ℓ¹, obtained by adjoining an identity element to the semigroup generated by two idempotents a and b subjected to the relation 0=abab ⋯ (length ℓ). We show that every monoid which generates a variety containing L₅¹ and is contained in the variety generated by L_ℓ¹ for some ℓ ≥ 5 is non-finitely based. We establish this result by analyzing τ-terms for M where τ is certain non-trivial congruence on the free semigroup, that is, we analyze words \bf u with the property that \bf u τ\bf v whenever M satisfies an identity \bf u ≈ \bf v. We also show that if τ is the trivial congruence on the free semigroup and ℓ ≤ 5 then the τ-terms (isoterms) for L_ℓ¹ carry no information about the non-finite basis property of L_ℓ¹.

Related

• Lee monoid L₄¹ is non-finitely based
• Non-finitely based monoids
• Finite semigroups embed in finitely presented congruence-free monoids
• Finite basis problems for stalactic, taiga, sylvester and Baxter monoids
• Catalan monoids inherently nonfinitely based relative to finite R-trivial semigroups