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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?

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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.

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