INVERSE FUNCTIONS OF NUMBER-THEORETIC FUNCTIONS(II)

If gf(x) =x for every x, then g is called a left inverse function of f and f is a right inverse function of g. If f is both left and right inverse function of g, then f and g are said to be mutually inverse to eaoh other. We show that (§1) the following results hold. A function f has a left inverse if and only if f is univalent, a function g has a right inverse if and only if g is exhaustive, i. e., g takes every (natural) number as values. Hence f has both left and right inverse if and only if f is both univalent and exhaustive, i. e., f is a permutation on the domain of natural numbers.Let g_1 and g_2 be two left inverse functions of the funoction f. If for every left inverse g of f, we have g_1(x)≤g(x)≤g_2(x), then g_1(x) is called the weak, and g_2(x) is the strong, left inverse function of f. Similarly we define the weak and the strong right inverse funotions.We show that (§2) every strict increasing function f must possess weak and strong left inverse funotions, and all of its left inverse functions must be exhaustive slow increasing (a function g(x) is slow increasing if and only if g(Sx)(?)Sg(x)=0, here s denotes the successor function). On the other hand, every exhaustive function g must possess weak and strong right inverse functions, and all of its right inverse functions must strict increasing.We show also that (§3):If f_1(x) and f_2(x) both take g(x) as their strong (weak) left inverse, then f_1(x)=f_2(x)(f_1(Sx)=f_2(Sx)).If g_1(x) and g_2(x) both take f(x) as their strong or weak right inverse, thenFrom these results we see that we may find a function from its strong (weak) left or right inverse function.Let there be f(c)≤f(c+1), then x(?)f(c) is called the f-residue of x and f(c+1)(?)x is the f-defioienoy of x.We show that (§4) if g(x) is any left inverse of f then |f(g(x))-x| must be either the f-residue or the f-deficiency of x.In the last section (§5) we consider the case of functions of several variables.