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Function S-rough sets are defined by R-function equivalence class, which have dynamic characteristic. Function S-rough sets have dynamic characteristic, law characteristic and law-hiding characteristic. Function S-rough sets can generate f-hiding law and $$ \bar f $$ -hiding law. By engrafting, crossing, and penetrating between the information security theory and function S-rough sets, the security hiding and the authentication of f-hiding law and $$ \bar f $$ -hiding law are given respectively in this paper. The fusion and share between function S-rough sets and information security theory is a new research direction of the application of information law in information system.

By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets (singular rough sets). S-rough sets are defined by element equivalence class that proposes dynamic characteristic. S-rough sets have dynamic characteristic. By introducing the function equivalence class (law equivalence class) that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets (function singular rough sets). Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.

Function S-rough sets, which was defined by R-function equivalence class, has dynamic characteristic and law characteristic., By combining dual of function one direction singular rough sets with intuitionistic fuzzy sets in this paper, the concepts such as adjoint intuitionistic fuzzy sets of law, degree of law variation and f-transfer coordination law were proposed; Based on adjoint intuitionistic fuzzy sets of law, the recognition theorem of law variation and the application of f transfer coordination law were given.

Two direction S-rough sets has two direction dynamic characteristics. By employing two direction S-rough sets, a generation method of two direction S-rough decision law is presented. The concepts of upper-decision law, lower-decision law, two direction S-rough decision law, two direction S-rough decision law kernel, two direction S-rough decision law band and two direction S-rough decision law hull are proposed. The theorems about decision law transfer and decision law mining are presented. The criterion of rough decision law mining is given.

Function S-rough sets is defined by function equivalence class, function is a kind of feature. By using of function S-rough sets, this paper gives a new feature selection method, gives concepts of lower approximation features and upper approximation features, furthermore, it presents the structure of F-feature pair, gives its characteristic discussion. F-feature selection method given in this paper has gotten applied in image processing, object recognition and etc, it is becoming a new research direction in recognition theory.

Function S-rough sets is defined by function equivalence class, function is a kind of law, law has heredity-variation characteristic, by using of this heredity- variation characteristic of function S-rough sets, this paper presents concept of f - heredity law, \(\overline{f}\) - variation law of image feature, gives image feature law generation theorem, and applications of image feature law heredity-variation in image hiding. All results of this paper have important application value in economic, military region.

Recognition of financial risk (investment risk and profit risk) has attracted more and more attention of investors, because each investor is threaten by the financial risk. Function S-rough set (function singular rough set) has law characteristic and the law has heredity characteristic. Using function S-rough set, this paper advances the recognition of financial risk law and gives its recognition model and an application example. Function S-rough set is defined by R-function equivalence class [u], ui∈[u] is a function (or a law). Function S-rough set is the general form of S-rough set (singular rough set) and S-rough set is the special case of function S-rough set. The results of this paper have lots of important applications