基于粗糙软集的(m, n)元(超)半环研究

The theories of rough sets, fuzzy sets and soft sets are mathematical tools for dealing with uncertainties. The fusion of these disciplines is one of hottest research topics in AI. It is well known that (m, n)-ary semirings and (m, n)-ary hypersemirings are representative (m, n)-ary structures and (m, n)-ary hyperstructures. Research on these structures is of significant value for developing the theories of semirings and hypersemirings. .In this project, by combining rough sets, soft sets and fuzzy sets, we will establish hybrid models of uncertainty processing and apply them to (m, n)-ary semirings and (m, n)-ary hypersemirings. This project mainly coveres the following topics: Two kinds of new soft rough fuzzy models and soft fuzzy rough models are established, some characterizations of upper and lower operators are discussed and uncertainty measurements are also given. Rough soft ideals and soft rough ideals of (m, n)-ary semirings are studied. At the same time, rough soft hyper ideals and soft rough hyper ideals of (m, n)-ary hypersemirings are also discussed. Four uncertainty models are explored in decision making fields in order to lay the theoretical foundation for corresponding uncertainty reasoning..Through the above research, we can fully reveal mathematics essence of rough soft set theory in order to promote rough soft sets in some applications of uncertainty decision making and uncertainty reasoning in artificial intelligence and information processing.

粗糙集、模糊集和软集理论均是处理不确定性问题的重要数学工具，其融合研究是人工智能领域的研究热点之一。(m, n)元半环与(m, n)元超半环是颇具代表性的(m, n)元代数和(m, n)元超代数结构，相关研究对于发展半环与超半环理论具有重要意义。.本项目将粗糙集、软集和模糊集相结合，构造不确定性处理的复合模型并将其应用于(m, n)元半环和(m, n)元超半环研究。主要包括：构造新的软粗糙模糊集和软模糊粗糙集模型，刻画模型中上下近似算子的特性与结构，给出概念不确定性度量方法；研究(m, n)元半环的粗糙软理想和软粗糙理想；刻画(m, n)元超半环的粗糙软超理想和软粗糙超理想；探讨四种不确定性模型在决策中的应用，为相应的不确定性推理研究奠定理论基础。.本项目研究将充分揭示粗糙软集理论的数学内涵，进一步推动粗糙软集理论在不确定性决策和不确定性推理等人工智能及信息处理领域中的应用。

粗糙集、模糊集和软集理论均是处理不确定性问题的重要数学工具，其融合研究是人工智能领域的研究热点之一。(m, n)元半环与(m, n)元超半环是颇具代表性的(m, n)元代数和(m, n)元超代数结构，相关研究对于发展半环与超半环理论具有重要意义。本项目将粗糙集、软集和模糊集相结合，构造不确定性处理的复合模型并将其应用于(m, n)元半环和(m, n)元超半环研究。主要包括：构造新的软粗糙模糊集和软模糊粗糙集模型，刻画模型中上下近似算子的特性与结构，给出概念不确定性度量方法；研究(m, n)元半环的粗糙软理想和软粗糙理想；刻画(m, n)元超半环的粗糙软超理想和软粗糙超理想；探讨四种不确定性模型在决策中的应用，为相应的不确定性推理研究奠定理论基础。本项目研究将充分揭示粗糙软集理论的数学内涵，进一步推动粗糙软集理论在不确定性决策和不确定性推理等人工智能及信息处理领域中的应用。