8 research outputs found
Proof Theory and Ordered Groups
Ordering theorems, characterizing when partial orders of a group extend to
total orders, are used to generate hypersequent calculi for varieties of
lattice-ordered groups (l-groups). These calculi are then used to provide new
proofs of theorems arising in the theory of ordered groups. More precisely: an
analytic calculus for abelian l-groups is generated using an ordering theorem
for abelian groups; a calculus is generated for l-groups and new decidability
proofs are obtained for the equational theory of this variety and extending
finite subsets of free groups to right orders; and a calculus for representable
l-groups is generated and a new proof is obtained that free groups are
orderable
Describing all bi-orderings on Thompson's group F
We describe all possible ways of bi-ordering Thompson group F: its space of
bi-orderings is made up of eight isolated points and four canonical copies of
the Cantor set.Comment: Final versio
Order, algebra, and structure: lattice-ordered groups and beyond
This thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics
On the dynamics of (left) orderable groups
We study left orderable groups by using dynamical methods. We apply these
techniques to study the space of orderings of these groups. We show for
instance that for the case of (non-Abelian) free groups, this space is
homeomorphic to the Cantor set. We also study the case of braid groups (for
which the space of orderings has isolated points but contains homeomorphic
copies of the Cantor set). To do this we introduce the notion of the Conradian
soul of an order as the maximal subgroup which is convex and restricted to
which the original ordering satisfies the so called conradian property, and we
elaborate on this notion.Comment: Final version, with updated references. Ann. Int. Fourier (to appear