On Compactification Lattices of Subsemigroups of SL(2,R)

Brigitte E. Breckner and Wolfgang A. F. Ruppert

Abstract: Using the tools introduced in [Breckner, B. E., and W. A. F. Ruppert, J. Lie Theory 11 (2001), 559--604], we investigate topological semigroup compactifications of closed connected submonoids with dense interior of Sl(2,R). In particular, we show that the growth of such a compactification is always contained in the minimal ideal, and describe the subspace of all minimal idempotents (typically a two-cell) and the maximal subgroups (these are always isomorphic with a compactification of R). For a large class of such semigroups we give explicit constructions yielding all possible topological semigroup compactifications and determine the structure of the compactification lattice.

Keywords: Bohr compactification, lattice of compactifications, asymptotic homomorphism, subsemigroups of Sl(2,R), Lie semigroups, Lie semialgebras, diamond product, rectangular domain, umbrella set, divisible semigroup, UDC semigroup

Classification (MSC2000): 22E15; 22E46, 22A15, 22A25

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