What type of algebra are Valiant's algebras for context-free grammars? In General Context-Free Recognition in Less than Cubic Time, Leslie Valiant gives an algorithm for transitive closure of matrices where the matrices are over an algebra $(S, \cup, \cdot, 0)$ where > the outer operation ($\cup$) is commutative and associative, ... the inner one ($\cdot$) distributes over it and ... $0$ is a multiplicative zero and an additive identity. Is there a name for such structures, which are generalisations of semirings?

Going by the terminology of nonassociative rings, we could call this a nonassociative semiring, and indeed Google gives some results for that term with a matching definition.