Weak Commutativity in Idempotent Semirings

Let U be the variety of idempotent semirings satisfying xy + yx = yx + xy. We give a description of the lattice L(U) of subvarieties of U: L(U) happens to be a 662-element distributive lattice which is isomorphic to a subdirect product of the lattices L(S+ l) and L(S. l), where L(S+ l) [L(S. l)] denotes the variety of all idempotent semirings whose additive [multiplicative] reduct is a semilattice. In particular, U = L(S+ l) ⋁ L(S. l). Every subvariety of U is finitely generated and finitely based. If S ∈ U, then both the additive reduct and the multiplicative reduct of S are regular bands. The structural relevance of the least U-congruence is investigated.