Semigroup structures for families of functions, III

Abstract

Let Fx and Fy denote families of subsets of the nonempty sets X and Y respectively and let be a function mapping Y onto X with the property that [H] ∈ Fx for each H∈Fy. Then the family l of all functions mapping X into Y such that [H] ∈Fy for each H ∈Fx is a semigroup if the product g of two such functions and g is defined by g = o o g (i.e., (Fg)(x) = (f(g(x))) for each x in X). With some restrictions on the families Fx and Fy and the function , these are the semigroups mentioned in the title and are the objects of investigation in this paper. The restrictions on fxfy and are sufficiently mild so that the semigroups considered here include such semigroups of functions on topological spaces as semi- groups of closed functions, semigroups of connected functions, etc.