| Abstract: | The H×H-Theorem. If S is a compact connected monoid with group of units H and with E(S) = {0,1}, and if S/(H×H) (the space of orbits HsH)
has a total order defning the quotient topology, then there is a one parameter semigroups I with E(I)=E(S) which commutes
elementwise with H. (In particular the function (h, i)→hi∶H×I→HI=S is a surmorphism, and S is cylindrical.) This is Theorem
VI in Elements of Compact Semigroups, by Hofmann and Mostert (p. 177). H. Carruth discovered a gap in the proof of this theorem
in 1971. The methods of proof presented here differ from theose originally suggested and do not use peripherality. byt do
use transformation group theory, and the authors' earlier results (Semigroup Forum 3 (1972), 31–42). The H×H-theorem is generalized
to yield a theorem which belongs to the context of Theorem VIII in the Elements (p. 204):Theorem: Let S be a compact monoid such that the orbit space S/(H×H) is a totally ordered connected space with M(S) as its minimal
point. If all regular D-classes are subsemigroups, then there is an I-semigroup with E(I)=E(S) which commutes elementwise
with H. (In particular S=HI as in the H×H theorem). The sufficient condition about the regular D-classes is clearly necessary.).
Further sample result:Theorem. IfH is a congruence in a compact connected monoid, with zero, then the centralizer of the group of units is connected. |
