We are concerned with
locally compact semitopological semigroups, with the variations for such semigroups
of the notions of left amenability and left thickness, and with systematizing the many
results which generalize a theorem of T. Mitchell for discrete semigroups: A subset T
of S is large enough to support a left-invariant mean on S if and only if T is left
thick; that is, for each finite subset F of S there is a v in S such that {fv∣f ∈ F} is
a subset of T.
In Part I: The textures of left thickness, we list many variations of left thickness
which have already been used, place them in a pattern of 90 = 5 × 3 × 3 × 2 such
conditions, and show that these fall into not more than six equivalence classes. In
Part II: The flavors of left-amenability, we list various kinds of amenability that have
already been used and try to match them with appropriate thickness conditions; that
is we try to find what thickness a set T in S must have to support a given kind of left
amenability, supposing, of course, that S itself supports that much amenability. In
this part we are also concerned with the thickness which a subsemigroup S′ of S
needs in order that some kind of amenability of S′ forces the same property on
S.
