The size of the set of left invariant means on an ELA semigroup

Abstract

Let S be an ELA semigroup and let m ( S ) \mathfrak {m}(S) be the smallest possible cardinality of the set { s ∈ S : F s = { s } } \{ s \in S:Fs = \{ s\} \} as F ranges over the finite subsets of S. The main purpose of this note is to show that if m ( S ) \mathfrak {m}(S) is infinite, then S has exactly 2 2 m ( S ) {2^{{2^{\mathfrak {m}(S)}}}} (multiplicative) left invariant means.