Vol. 101, No. 1, 1982

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Left thick to left lumpy—a guided tour

Mahlon M. Day

Vol. 101 (1982), No. 1, 71–92

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 {fvf 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 Sof S needs in order that some kind of amenability of Sforces the same property on S.

Mathematical Subject Classification 2000
Primary: 43A07
Secondary: 22A20, 54H15
Received: 19 July 1979
Published: 1 July 1982
Mahlon M. Day