Geodesic
Green's $\mathcal{D}$-relation for the multiplicative reduct of an idempotent semiring
Archivum mathematicum, Tome 36 (2000) no. 2, pp. 77-93.

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The idempotent semirings for which Green’s ${\cal D}$-relation on the multiplicative reduct is a congruence relation form a subvariety of the variety of all idempotent semirings. This variety contains the variety consisting of all the idempotent semirings which do not contain a two-element monobisemilattice as a subsemiring. Various characterizations will be given for the idempotent semirings for which the ${\cal D}$-relation on the multiplicative reduct is the least lattice congruence.
Classification : 16Y60, 20M10
Keywords: idempotent semiring; variety; Green relations; band; bisemilattice 
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     title = {Green's $\mathcal{D}$-relation for the multiplicative reduct of an idempotent semiring},
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Pastijn, F.; Zhao, Xianzhong. Green's $\mathcal{D}$-relation for the multiplicative reduct of an idempotent semiring. Archivum mathematicum, Tome 36 (2000) no. 2, pp. 77-93. http://geodesic.mathdoc.fr/item/ARM_2000__36_2_a0/