The continuity in the inductive sets: continuity of superposition and related results (Q2703312)

The author studies the problem of closure of monotone and continuous function classes with respect to superposition and monotonicity and continuity of superposition itself. In particular, the following results are proved. Functions \(f_1:D\to D_1,\ldots, f_{m}:D\to D_{m}\) are monotone (continuous) if and only if the function \(\chi^{(m)}(f_1,\ldots,f_{m}):D\to D_1\times\ldots\times D_{m}\) is monotone (continuous). Here \(D,D_1,\ldots,D_{m}\) are inductive sets, \(\chi^{(m)}\) is the operation (called collection) \(\chi^{(m)}:(D\to D_1)\times\ldots\times (D\to D_{m})\to(D\to D_1\times\ldots\times D_{m})\), \(m=2,3,\ldots\), such that \(\chi^{(m)} (f_1,\ldots,f_{m})(x)= \langle f_1(x),\ldots,f_{m}(x)\rangle\) for the corresponding functions \(f_1,\ldots,f_{m}\) and \(x\in D\). Superposition \(S^{(m+1)}\) has a representation \(S^{(m+1)}(g,f_1,\ldots, f_{m})=\chi^{(m)}(f_1,\ldots,f_{m})\circ g\). Using the fact that the class of monotone (continuous) functions is closed with respect to products the author proves the closure of this class with respect to superpositions.