13 research outputs found
Nonequational Stable Groups
We introduce a combinatorial criterion for verifying whether a formula is not
the conjunction of an equation and a co-equation. Using this, we give a
transparent proof for the nonequationality of the free group, which was
originally proved by Sela. Furthermore, we extend this result to arbitrary free
products of groups (except ), providing an abundance
of new stable nonequational theories.Comment: 10 pages, 1 figur
Comparing axiomatizations of free pseudospaces
Independently and pursuing different aims, Hrushovski and Srour (On stable non-equational theories. Unpublished manuscript, 1989) and Baudisch and Pillay (J Symb Log 65(1):443–460, 2000) have introduced two free pseudospaces that generalize the well know concept of Lachlan’s free pseudoplane. In this paper we investigate the relationship between these free pseudospaces, proving in particular, that the pseudospace of Baudisch and Pillay is a reduct of the pseudospace of Hrushovski and Srour
A model theoretic study of right-angled buildings
We study the model theory of countable right-angled buildings with infinite
residues. For every Coxeter graph we obtain a complete theory with a natural
axiomatisation, which is -stable and equational. Furthermore, we
provide sharp lower and upper bounds for its degree of ampleness, computed
exclusively in terms of the associated Coxeter graph. This generalises and
provides an alternative treatment of the free pseudospace.Comment: A number of small typos found by typesetter correcte
PAC structures as invariants of finite group actions
We study model theory of actions of finite groups on substructures of a
stable structure. We give an abstract description of existentially closed
actions as above in terms of invariants and PAC structures. We show that if the
corresponding PAC property is first order, then the theory of such actions has
a model companion. Then, we analyze some particular theories of interest
(mostly various theories of fields of positive characteristic) and show that in
all the cases considered the PAC property is first order
Stable domination and independence in algebraically closed valued fields
We seek to create tools for a model-theoretic analysis of types in
algebraically closed valued fields (ACVF). We give evidence to show that a
notion of 'domination by stable part' plays a key role. In Part A, we develop a
general theory of stably dominated types, showing they enjoy an excellent
independence theory, as well as a theory of definable types and germs of
definable functions. In Part B, we show that the general theory applies to
ACVF. Over a sufficiently rich base, we show that every type is stably
dominated over its image in the value group. For invariant types over any base,
stable domination coincides with a natural notion of `orthogonality to the
value group'. We also investigate other notions of independence, and show that
they all agree, and are well-behaved, for stably dominated types. One of these
is used to show that every type extends to an invariant type; definable types
are dense. Much of this work requires the use of imaginary elements. We also
show existence of prime models over reasonable bases, possibly including
imaginaries