Abstract
We call a semigroup on which the Green’s quasiorder \(\le _{{\mathcal {J}}}\) (\(\le _{{\mathcal {L}}}\), \(\le _{{\mathcal {R}}}\)) is operation-compatible, a \(\le _{{\mathcal {J}}}\)-compatible (\(\le _{{\mathcal {L}}}\)-compatible, \(\le _{{\mathcal {R}}}\)-compatible) semigroup. We study the classes of \(\le _{{\mathcal {J}}}\)-compatible, \(\le _{{\mathcal {L}}}\)-compatible and \(\le _{{\mathcal {R}}}\)-compatible semigroups, using the smallest operation-compatible quasiorders containing Green’s quasiorders as a tool. We prove a number of results, including the following. The class of \(\le _{{\mathcal {L}}}\)-compatible (\(\le _{{\mathcal {R}}}\)-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is \(\le _{{\mathcal {J}}}\)-compatible if and only if it is a semilattice of simple semigroups. Every negatively orderable semigroup can be embedded into a negatively orderable \(\le _{{\mathcal {J}}}\)-compatible semigroup.
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Communicated by Jean-Eric Pin.
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Juhász, Z., Vernitski, A. Semigroups with operation-compatible Green’s quasiorders. Semigroup Forum 93, 387–402 (2016). https://doi.org/10.1007/s00233-016-9792-y
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DOI: https://doi.org/10.1007/s00233-016-9792-y