\(\mathcal L\)-subvarieties of the variety of idempotent semirings.
A semiring \((S;+,\cdot )\) is an algebra of type \((2,2)\) where \((S,+), \, (S,\cdot )\) are semigroups and two distributive identities, \(x\cdot (y+z) = x\cdot y + x\cdot z, \, (x+y)\cdot z = x\cdot z + y\cdot z\), are satisfied. The authors study the Green relations \(\mathcal D\) and \(\mathcal L\) on the semigroups \((S,+), \, (S,\cdot )\) for a semiring \((S;+,\cdot )\). In the case of idempotent semirings, they classify all the varieties where \(\mathcal D\) and/or \(\mathcal L\) for \((S,+)\) and/or \((S,\cdot )\) are congruences or minimal or maximal congruences. They derive equational bases for all of these possibilities.
scientific article; zbMATH DE number 2158733