On quotients of ω* and automorphisms of P(ω)/fin that preserve or invert the shift

Abstract

The first part of the dissertation concerns the space ω*:=βω\ω and its quotients. A 2002 result from Bella, Dow, Hart, Hrusak, van Mill and Ursino implies that homeomorphisms between 0-dimensional second-countable Hausdorff quotients of ω* can always be lifted to self-homeomorphisms of ω*. We show that 0-dimensionality can be dropped and also prove that if the map between the quotients is continuous, but not necessarily a homeomorphism, then it can be lifted to a continuous map from ω* into itself. We prove that all quotient maps from ω* onto products of compact metrizable spaces are restrictions of quotient maps from βω (with the same range). We then defend our choice of hypotheses for this last result by showing that there is a quotient map from ω* onto the double arrow space (which is separable, 0-dimensional, first-countable, compact and Hausdorff) with no continuous extension from βω.<br /> In the second part, the focus is on the Boolean algebra <b>P</b>(ω)/<b>fin</b> and the automorphism<i>s</i> of <b>P</b>(ω)/<b>fin</b> called <i>the shift</i>, which is induced by the map <i>n</i> → <i>n</i>+1 in ω. We show that the automorphisms <i>s^m</i> for <i>m</i> ∈ℤ are the only trivial automorphisms of <b>P</b>(ω)/<b>fin</b> which commute with the shift. Together with a 1993 result of Velickovic, this characterizes all automorphisms of (<b>P</b>(ω)/<b>fin</b>, <i>s</i>) under <b>OCA</b>+<b>MA</b>_ℵ₁. We study the algebra <b>Per</b> of all elements of <b>P</b>(ω)/<b>fin</b> with finite orbit under the action of the shift, and characterize all the automorphisms of <b>Per</b> which commute with the shift (many of which are not powers of the shift) or conjugate the shift to its inverse. Then, we show that every automorphism of the group of trivial automorphisms of <b>P</b>(ω)/<b>fin</b> either preserves or inverts (additively) the index function introduced by van Douwen in 1990. We also construct a set of 2^א₀ many elements of <b>P</b>(ω)/<b>fin</b> such that the substructures of (<b>P</b>(ω)/<b>fin</b>, <i>s</i>) generated by each of these elements individually are pairwise isomorphic, and a set of 2^א₀ many elements of <b>P</b>(ω)/<b>fin</b> such that the substructures of (<b>P</b>(ω)/<b>fin</b>, <i>s</i>) generated by each of these elements individually are pairwise non-isomorphic. We finish with a method for constructing automorphisms of <b>P</b>(ω)/<b>fin</b> which conjugate the shift to itself or to its inverse on any given countable subalgebra.