22,107 research outputs found
Elementary quotient completion
We extend the notion of exact completion on a weakly lex category to
elementary doctrines. We show how any such doctrine admits an elementary
quotient completion, which freely adds effective quotients and extensional
equality. We note that the elementary quotient completion can be obtained as
the composite of two free constructions: one adds effective quotients, and the
other forces extensionality of maps. We also prove that each construction
preserves comprehensions
Algebra and logic. Some problems
The paper has a form of a talk on the given topic. It consists of three
parts.
The first part of the paper contains main notions, the second one is devoted
to logical geometry, the third part describes types and isotypeness. The
problems are distributed in the corresponding parts. The whole material
oriented towards universal algebraic geometry (UAG), i.e., geometry in an
arbitrary variety of algebras . We consider logical geometry (LG) as a
part of UAG. This theory is strongly influenced by model theory and ideas of
A.Tarski and A.I.Malcev
A co-free construction for elementary doctrines
We provide a co-free construction which adds elementary structure to a
primary doctrine. We show that the construction preserves comprehensions and
all the logical operations which are in the starting doctrine, in the sense
that it maps a first order many-sorted theory into a the same theory formulated
with equality. As a corollary it forces an implicational doctrine to have an
extentional entailment
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
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