Abstract
We prove first that every mathcal {H}-commutative semigroup is stable. Using this result [and some results from the standard text (Nagy, Special classes of semigroups, Kluwer, Dordrecht, 2001)], we give two equivalent conditions for a semigroup to be an archimedean mathcal {H}-commutative semigroup containing an idempotent element. It turns out that this result can be partially extended to mathcal {L}-commutative semigroups and quasi-commutative semigroups.
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