The congruences on S(X) which commute with V (Q579422)

Let S be any semigroup and let V be the relation on S defined by \(V=\{(a,b)\in S\times S:\) a and b are mutually inverse to one another\(\}\). \textit{R. J. Koch} and \textit{B. L. Madison} [Simon Steven 57, 273-283 (1983; Zbl 0525.20044)] have posed the problem of describing all those congruences on S which commute with the relation V. We solve this problem for S(X), the semigroup of all continuous selfmaps of the topological space X. It turns out that there are exactly two such congruences if X is connected and exactly six if X is not connected.