坡是两个元素的乘积小于等于每个因子的加法幂等半环。布尔代数、模糊代数、分配格和热带半环是坡的典型例子。本研究报告包含四个主要部分。第一部分，建立坡的理想和其商坡的理想之间成立的一个对应关系, 研究一般坡的代数性质。第二部分，证明在交换坡上n阶矩阵的序指数至多为(n-1)^2+1且这是确界，从而解决坡矩阵理论中一个公开问题。第三部分，给出坡矩阵理论在坡值模糊双向联想记忆网络动态分析中的应用。第四部分，求解任意Brouwer格(一种特殊的交换剩余格序坡)上矩阵方程，从而解决模糊关系方程理论中一个未解决的问题。

Inclines are additively idempotent semirings in which products are less than or equal to either factor. Boolean algebras, fuzzy algebras, distributive lattices and tropical semirings are prototypical examples of inclines.The present report consists of four main parts.The first part establishes a correspondence relation between ideals of an incline and ideals of its quotient incline, and studies algebraic properties of general inclines.The second part solves an open problem in the incline matrix theory by showing that theorder-index of an nxn matrix over a commutative incline is at most (n-1)^2+1, and this bound is sharp.The third part presents an application of the incline matrix theory in the dynamic analysis ofincline-valued fuzzy bidirectional associative memories.The fourth part solves an unsolved question in the theory of fuzzy relation equations by resolvinga matrix equation over an arbitrary Brouwerian lattice which is a special kind of commutativeresiduated lattice-ordered inclines.