Category of grey sets (Q6637452)

Fuzzy sets, rough sets and grey systems provide three different but overlapping models for the representation of uncertainties in sets. A grey set combines vagueness and incompleteness into one model and can be specified to interval-valued fuzzy sets or rough sets under special situations. However, a grey set can represent situations not covered by interval-valued fuzzy sets or rough sets. In grey systems, the information is classified into three categories: white with completely certain information, grey with insufficient information, and black with totally unknown information. In this context, grey numbers are the basic concepts. A grey number is defined as a number with clear upper and lower boundaries but which has an unknown position within the boundaries. This definition for the system is expressed mathematically as \(g^{\pm} \in [g^{-}, g^{+}]=\{g^{-} \leq t \leq g^{+}\}\), where \(g^{\pm}\) is a grey number, \(t\) is information, \(g^{-}\), and \(g^{+}\) are the upper and lower limits of the information. Similar to grey numbers, sets are classified into three different categories: Black sets, Grey Sets, and White sets. Let \(U\) denote a universe of discourse. Given a set \(A \subseteq U\), if the characteristic function value of \(x\) with respect to \(A\) can be expressed with a grey number \(g^{\pm}_A(x) \in \bigcup_1^n [a_i^{-}, a_i^{+}] \in D[0, 1]^{\pm}\) \N\[ \chi_A : U \to D[0, 1]^{\pm} \] \Nthen \(A\) is a grey set. Here \(D[0, 1]^{\pm}\) denotes the set of all grey numbers within the interval \([0, 1]\). \N\NIn the paper under review, the authors define a new category denoted by \textbf{Gset} in which the objects are grey sets, and morphisms are grey functions defined as follows. Let \(X=(U, \chi_A)\) and \(Y=(V, \chi_B)\) be two grey sets, with \(\chi_A : U \to D[0, 1]^{\pm}\) and \(\chi_B : V \to D[0, 1]^{\pm}\). A grey morphism between grey sets \(X\) and \(Y\) is a function \(f : U \to V\) such that (upper) lower \(\chi_A(x) \leq \) (upper) lower \(\chi_Bf(x)\) or shorter \(\chi^{\pm}_A(x) \leq \chi^{\pm}_Bf(x)\). Some basic properties of this category are investigated.