Amenable transformation semigroups

Abstract

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.

Characterization of finite amenable transformation semigroups

“…The notion of an S-invariant mean was introduced and a number of basic properties developed in [3]. Important special cases of z-semigroups occur when X is in an abstract semigroup and S either the semigroup of transformations of X induced by left multiplication or that induced by right multiplication; these will be called /-semigroup and r-semigroups, respectively.…”

Characterization of finite amenable transformation semigroups: corrigendum

“…Theorem 6 of Wilde (1971) is in error. F</>lner's condition is necessary, but not sufficient, for amenability of a T-semigroup.…”

“…F</>lner's condition is necessary, but not sufficient, for amenability of a T-semigroup. For clearly sufficiency would imply that every finite T-semigroup is amenable, and the simple example on p. 503 of Wilde (1971) shows that this is not so. We refer the reader to Namioka (1964) for a lucid explanation of the fundamental ideas involved.…”

Computation of the transformation semigroups on three letters

“…In [1 ] the notion of an invariant mean for general r-semigroups was formulated and some basic results established. In connection with that study, the problem of computing all the T-semigroups on three letters arose; it was felt that this collection would provide insight and examples as a basis for interesting further studies.…”