Abstract. We show that there are uncountably many varieties of semirings between the variety generated by ${\bf Rel}(2)$, the semiring of binary relations on a two-element set, and the variety generated by its particular subsemiring ${\bf\Sigma }_7$ (which is proved to be nonfinitely based in an earlier paper, as a first instance of such kind). An analogous result is obtained for semigroup varieties. As a consequence, we are able to exhibit a pseudorecursive variety of additively idempotent semirings.

AMS Subject Classification (1991): 16Y60, 20M07, 08B05

Received February 12, 2007, and in revised form June 20, 2007. (Registered under 6004/2009.)