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Green’s relations and left amenable semigroups
Published online by Cambridge University Press: 15 September 2025
Abstract
In this note, some conditions are investigated under which the left amenability of a semigroup S is a consequence of the left amenability of its subsemigroups. It is known that for the Green’s relation 
$\mathcal {H}^S$ on S, an 
$\mathcal {H}^S$-class of S is a semigroup if and only if it is a subgroup of S, and hence it contains a unique identity. Let S be a semigroup such that every 
$\mathcal {H}^S$-class of S is a group and E, the set of idempotents of S, is a subsemigroup of S. As the main result of this note, applying the above fact, a connection between left amenability of S, left amenability of E, and left amenability of its 
$\mathcal {H}^S$-classes is established.
As an application, I completely determine left amenable Clifford semigroups and left amenable rectangular groups, when they are left amenable with some measure such that the union of every collection of 
$\mathcal {H}^S$-classes of S with zero measure has zero measure (especially, when E is finite or when E is countable and it is left amenable with a measure which is countably additive). Indeed, I show that under this assumption, (i) a Clifford semigroup S is left amenable if and only if E has a zero element z and 
$H_z$, the 
$\mathcal {H}^S$-class of S which contains z, is a left amenable group and (ii) a rectangular group S is left amenable if and only if it is a right group and its 
$\mathcal {H}^S$-classes are left amenable groups.
MSC classification
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
This work was supported by the Iranian National Science Foundation (INSF) Grant No. 98021433.
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