Generalized matroids based on three-way decision models

https://doi.org/10.1016/j.ijar.2017.07.012Get rights and content
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Highlights

  • This paper introduces two generalizations of matroids based on subset-evaluation based three-way decision models.

  • Two explanations of three-way fuzzy matroids are given.

  • Three-way matroids and three-way fuzzy matroids provide new context for some famous algorithms in matroid theory.

Abstract

Three-way decision theory is an extension of the commonly used binary-decision model with an added third option. It is originally introduced to explain the three regions of probabilistic rough sets. Every object in a three-way decision model can be assigned to one of the three regions according to its evaluation value under an evaluation function. This paper first introduces three-way decision models based on subset-evaluation which generalize the original models. By the axiomatic approach, we characterize a matroid in terms of evaluation function and then define three-way matroids based on this characterization. Furthermore, three-way matroids are generalized to three-way fuzzy matroids and an equivalent description of three-way fuzzy matroid in terms of fuzzy independent set system is presented. Finally, we give the second description of three-way fuzzy matroid: a three-way fuzzy matroid is exactly the greatest element of an equivalence class. Additionally, relations of notions introduced in this paper are also pointed out.

Keywords

Rough sets
Three-way decisions
Three-way matroids
Three-way fuzzy matroids

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This work was partly supported by the National Natural Science Foundation of China (61202178, 61472471, 71161016), the Natural Science Foundation of Shaanxi Province of China (2017JM7022, 2017JM1036) and the Fundamental Research Funds for the Central Universities (JB170702, JBZ170601).