Abstract
Ideas of three-way decisions proposed by Yao come from rough sets. It is well known that there are three basic elements in three-way decisions theory, which are ordered set as to define three regions, object set contained in evaluation function and evaluation function to make three-way decisions. In this paper these three basic elements are called decision measurement, decision condition and evaluation function, respectively. In connection with the three basic elements this paper completes three aspects of work. The first one is to introduce axiomatic definitions for decision measurement, decision condition and evaluation function; the second is to establish three-way decisions space; and the third is to give a variety of three-way decisions on three-way decisions spaces. Existing three-way decisions are the special examples of three-way decisions spaces defined in this paper, such as three-way decisions based on fuzzy sets, random sets and rough sets etc. At the same time, multi-granulation three-way decisions space and its corresponding multi-granulation three-way decisions are also established. Finally this paper introduces novel dynamic two-way decisions and dynamic three-way decisions based on three-way decisions spaces and three-way decisions with a pair of evaluation functions.
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