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For any set X denote by m(X) the Banach space of all bounded real-valued functions on X, equipped with the supremum norm, and denote by (X) the semigroup (under functional composition) of all transformations of X, i.e. mappings with domain X and range contained in X. A pair (X, S), where S is a subsernigroup of (X), will be called a transformation semigroup. Important examples are obtained by letting X be the underlying set in an abstract semigroup and considering the pairs (X, S1) and (X, S2), where S1 [Sn] denotes the set of left [right] multiplication mappings of X. We shall call transformation semigroups in these classes of examples l-[r-] semigroups.

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