Representations of certain compact semigroups by 𝐻𝐿-semigroups

Abstract

An <italic>HL</italic>-semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into <italic>HL</italic>-semigroups to separate points of <italic>S</italic>; or equivalently, is <italic>S</italic> isomorphic to a strict projective limit of <italic>HL</italic>-semigroups? An affirmative answer is given in the case that <italic>S</italic> is an irreducible semigroup. If <italic>S</italic> is irreducible and separable, it is shown that <italic>S</italic> admits enough homomorphisms into finite dimensional <italic>HL</italic>-semigroups to separate points of <italic>S</italic>.

Locations

• Transactions of the American Mathematical Society - View - PDF