Local finiteness for Green’s relations in semigroup varieties

ABSTRACT

A semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ℋ, ℛ, ℒ, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.

KEYWORDS:

• Green’s relation
• local finiteness
• Novikov–Adian group
• semigroup variety

MATHEMATICS SUBJECT CLASSIFICATION:

• 20M07
• 20M10
• 20M12
• 20F50

Acknowledgment

The authors thank the anonymous referee for a number of useful remarks.

Notes

¹Here and throughout we use expressions like A: = B to emphasize that A is defined to be B.

²There is a misprint in the formulation of this result in [37]: “left ideal” should read “right ideal”.

³Observe that we are not in a position to apply the induction assumption to S∕T.

Additional information

Funding

Mikhail Volkov acknowledges support from the Russian Foundation for Basic Research, project no. 17-01-00551, the Ministry of Education and Science of the Russian Federation, project no. 1.3253.2017, and the Competitiveness Program of Ural Federal University. Pedro V. Silva was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020. Filipa Soares was partially supported by CEMAT-Ciências (UID/Multi/04621/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020.