The finite basis problem for $2\times 2$ triangular Boolean matrix semirings and incidence semirings

An explicit finite identity basis is given for the eight-element semiring of upper triangular Boolean \(2\times2\) matrices in the signature \((+,\cdot)\). The basis consists of a known multiplicative basis, the ai-semiring laws, and 30 mixed identities, each using at most eight variables. The proof