Green’s relations and left amenable semigroups
Behnam Khosravi · Canadian Mathematical Bulletin · 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 upper H Superscript upper S $\mathcal {H}^S$ H S on S , an upper H Superscript upper S $\mathcal {H}^S$ 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 upper H Superscript upper S $\mathcal {H}^S$ 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 upper H Superscript upper S $\mathcal {H}^S$ 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 upper H Superscript upper S $\mathcal {H}^S$ 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 upper H Subscript z $H_z$ H z , the