Order-indices and order-periods of 3x3 matrices over commutative inclines

TL;DR

This paper proves that for all 3x3 matrices over any commutative incline, the 11th power is less than or equal to the 5th power, resolving an open problem in incline algebra.

Contribution

It establishes a universal inequality for 3x3 matrices over commutative inclines, extending previous partial results and solving a longstanding open problem.

Findings

Proves A^{11} ≤ A^{5} for all 3x3 matrices over commutative inclines.

Uses prime numbers to establish the inequality.

Answers an open problem posed in 1984.

Abstract

An incline is an additively idempotent semiring in which the product of two elements is always less than or equal to either factor. By making use of prime numbers, this paper proves that A^{11} is less than or equal to A^5 for all 3x3 matrices A over an arbitrary commutative incline, thus giving an answer to an open problem "For 3x3 matrices over any incline (even noncommutative) is X^5 greater than or equal to X^{11}?", proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984.