All solid varieties of semirings (Q1604379)

A hypersubstitution results from a substitution of each operation symbol by a term of the same arity. A variety \(V\) is called solid if for every identity \(s\approx t\) which holds in \(V\) and every hypersubstitution, the identity which is obtained from \(s\approx t\) by applying the hypersubstitution is again satisfied in \(V\). A semiring \(S\) is called a normal idempotent distributive semiring if the reducts \((S,+)\) and \((S,.)\) are normal bands and also the dual distributive laws \(x+yz=(x+y)(x+z)\) and \(xy+z=(x+z)(y+z)\) hold. It is found that there are exactly three nontrivial solid varieties of semirings, namely the variety of all normal idempotent distributive semirings and two of its nontrivial subvarieties.