58 research outputs found
Partial Automorphism Semigroups
We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism (elementary equivalence) of the subsemigroups yields isomorphism (elementary equivalence) of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a computable structure, up to computable isomorphism, from the isomorphism type of its inverse semigroup of computable partial automorphisms
On cohesive powers of linear orders
Cohesive powers of computable structures are effective analogs of
ultrapowers, where cohesive sets play the role of ultrafilters. Let ,
, and denote the respective order-types of the natural numbers,
the integers, and the rationals when thought of as linear orders. We
investigate the cohesive powers of computable linear orders, with special
emphasis on computable copies of . If is a computable
copy of that is computably isomorphic to the standard presentation of
, then every cohesive power of has order-type . However, there are computable copies of , necessarily not
computably isomorphic to the standard presentation, having cohesive powers not
elementarily equivalent to . For example, we show that
there is a computable copy of with a cohesive power of order-type
. Our most general result is that if is either a set or a set, thought of as a
set of finite order-types, then there is a computable copy of with a
cohesive power of order-type , where denotes
the shuffle of the order-types in and the order-type . Furthermore, if is finite and non-empty, then there is a
computable copy of with a cohesive power of order-type
Trivial, Strongly Minimal Theories Are Model Complete After Naming Constants
We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(MM) is a model complete LM-theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0 -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ05
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