302 research outputs found
Expressive Power in First Order Topology
A first order representation (fo.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions one f.o.r. is at least as expressive as another relative to a class of spaces and one class of spaces is definable in another relative to an f.o.r. , and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positiveuniversal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Characterizations of categories of commutative C*-subalgebras
We aim to characterize the category of injective *-homomorphisms between
commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to
finding a weakly terminal commutative subalgebra of A, and solve the latter for
various C*-algebras, including all commutative ones and all type I von Neumann
algebras. This addresses a natural generalization of the Mackey-Piron
programme: which lattices are those of closed subspaces of Hilbert space? We
also discuss the way this categorified generalization differs from the original
question.Comment: 24 page
Singly generated quasivarieties and residuated structures
A quasivariety K of algebras has the joint embedding property (JEP) iff it is
generated by a single algebra A. It is structurally complete iff the free
countably generated algebra in K can serve as A. A consequence of this demand,
called "passive structural completeness" (PSC), is that the nontrivial members
of K all satisfy the same existential positive sentences. We prove that if K is
PSC then it still has the JEP, and if it has the JEP and its nontrivial members
lack trivial subalgebras, then its relatively simple members all belong to the
universal class generated by one of them. Under these conditions, if K is
relatively semisimple then it is generated by one K-simple algebra. It is a
minimal quasivariety if, moreover, it is PSC but fails to unify some finite set
of equations. We also prove that a quasivariety of finite type, with a finite
nontrivial member, is PSC iff its nontrivial members have a common retract. The
theory is then applied to the variety of De Morgan monoids, where we isolate
the sub(quasi)varieties that are PSC and those that have the JEP, while
throwing fresh light on those that are structurally complete. The results
illuminate the extension lattices of intuitionistic and relevance logics
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
A Category of Ordered Algebras Equivalent to the Category of Multialgebras
It is well known that there is a correspondence between sets and complete,
atomic Boolean algebras (CABA's) taking a set to its power-set and,
reciprocally, a complete, atomic Boolean algebra to its set of atomic elements.
Of course, such a correspondence induces an equivalence between the opposite
category of and the category of CABA's. We extend this result by
taking multialgebras over a signature , specifically those whose
non-deterministic operations cannot return the empty-set, to CABA's with their
zero element removed and a structure of -algebra compatible with its
order; reciprocally, one of these "almost Boolean" -algebras is taken
to its set of atomic elements equipped with a structure of multialgebra over
. This leads to an equivalence between the category of
-multialgebras and a category of ordered -algebras. The
intuition, here, is that if one wishes to do so, non-determinism may be
replaced by a sufficiently rich ordering of the underlying structures
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