Idempotent Triangular Matrices over Additively Idempotent Semirings: Decompositions into Products of Semicentral Idempotents

The explicit forms of idempotent and semicentral idempotent triangular matrices over an additively idempotent semiring are obtained. We define a diamond composition of idempotents and give a representation of an idempotent n×n matrix as an (n−1)th degree of a sum of diamond compositions of semicentral idempotents. We construct a decomposition of a strictly upper matrix, a unitriangular matrix, and a nil-clean matrix by semicentral idempotents.