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Keywords:
idempotent semiring; variety; Green relations; band; bisemilattice
idempotent semiring; variety; Green relations; band; bisemilattice
Summary:
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.
References:
[1] Howie J. M.: Fundamentals of Semigroup Theory. Oxford Science Publications, Oxford, 1995. MR 1455373 | Zbl 0835.20077
[2] McKenzie R., and A. Romanowska: Varieties of $\cdot $-distributive bisemilattices. Contributions to General Algebra, (Proc. Klagenfurt Conf., Klagenfurt 1978), 213–218, Heyn, Klagenfurt, 1979. MR 0537422
[3] Pastijn F., and Y. Q. Guo: The lattice of idempotent distributive semiring varieties. Science in China (Series A) 42 (8) (1999), 785–804. MR 1738550
[4] Sen M. K., Guo Y. Q., and K. P. Shum: A class of idempotent semirings. preprint. MR 1828821
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