Maximum order-index of matrices over commutative inclines: an answer to an open problem

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)2 + 1. This is an answer to an open problem “Compute the maximum order-index of a member of Mn(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where Mn(L) is the set of all n × n matrices over an incline L.