1,358 research outputs found
A Note on the Set-Theoretic Representation of Arbitrary Lattices
Every lattice is isomorphic to a lattice whose elements are sets of sets, and
whose operations are intersection and an operation extending the union of two
sets of sets A and B by the set of all sets in which the intersection of an
element of A and of an element of B is included. This representation spells out
precisely Birkhoff's and Frinks's representation of arbitrary lattices, which
is related to Stone's set-theoretic representation of distributive lattices.Comment: 3 pages (published in A. Krapez ed., A Tribute to S.B. Presic,
Mathematical Institute, Belgrade, 2001, pp. 99-101
Identity of Proofs Based on Normalization and Generality
In general proof theory there are two approaches to the question of identity
criteria for proofs. The first approach, which stems from Prawitz, Kreisel and
Lambek, and is based on normalization of proofs, gives good results in
intuitionistic, but not in classical logic. The second approach, which stems
from Lambek, Mac Lane and Kelly, and is inspired by the generality of proofs,
seems to be more promissing in classical logic.Comment: 29 pages, corrected remark about difunctionality, updated reference
Abstraction and Application in Adjunction
The postulates of comprehension and extensionality in set theory are based on
an inversion principle connecting set-theoretic abstraction and the property of
having a member. An exactly analogous inversion principle connects functional
abstraction and application to an argument in the postulates of the lambda
calculus. Such an inversion principle arises also in two adjoint situations
involving a cartesian closed category and its polynomial extension. Composing
these two adjunctions, which stem from the deduction theorem of logic, produces
the adjunction connecting product and exponentiation, i.e. conjunction and
implication.Comment: 15 page
On Sets of Premises
Conceiving of premises as collected into sets or multisets, instead of
sequences, may lead to triviality for classical and intuitionistic logic in
general proof theory, where we investigate identity of deductions. Any two
deductions with the same premises and the same conclusions become equal. In
terms of categorial proof theory, this is a consequence of a simple fact
concerning adjunction with a full and faithful functor applied to the
adjunction between the diagonal functor and the product biendofunctor, which
corresponds to the conjunction connective.Comment: 13 page
Simplicial Endomorphisms
The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in
the simplicial category, are submonoids of monoids one finds in Temperley-Lieb
algebras, and as the monoids of Temperley-Lieb algebras are linked to
situations where an endofunctor is adjoint to itself, so the monoids of
simplicial endomorphisms are linked to arbitrary adjoint situations. This link
is established through diagrams of the kind found in Temperley-Lieb algebras.
Results about these matters, which were previously prefigured up to a point,
are here surveyed and reworked. A presentation of monoids of simplicial
endomorphisms by generators and relations has been given a long time ago. Here
a closely related presentation is given, with completeness proved in a new and
self-contained manner.Comment: 33 pages, new version, updated reference
Medial Commutativity
It is shown that all the assumptions for symmetric monoidal categories flow
out of a unifying principle involving natural isomorphisms of the type
, called
medial commutativity. Medial commutativity in the presence of the unit object
enables us to define associativity and commutativity natural isomorphisms. In
particular, Mac Lane's pentagonal and hexagonal coherence conditions for
associativity and commutativity are derived from the preservation up to a
natural isomorphism of medial commutativity by the biendofunctor .
This preservation boils down to an isomorphic representation of the Yang-Baxter
equation of symmetric and braid groups. The assumptions of monoidal categories,
and in particular Mac Lane's pentagonal coherence condition, are explained in
the absence of commutativity, and also of the unit object, by a similar
preservation of associativity by the biendofunctor . In the final
section one finds coherence conditions for medial commutativity in the absence
of the unit object. These conditions are obtained by taking the direct product
of the symmetric groups for .Comment: 27 pages, minor addition
Representing Conjunctive Deductions by Disjunctive Deductions
A skeleton of the category with finite coproducts D freely generated by a
single object has a subcategory isomorphic to a skeleton of the category with
finite products C freely generated by a countable set of objects. As a
consequence, we obtain that D has a subcategory equivalent with C. From a
proof-theoretical point of view, this means that up to some identifications of
formulae the deductions of pure conjunctive logic with a countable set of
propositional letters can be represented by deductions in pure disjunctive
logic with just one propositional letter. By taking opposite categories, one
can replace coproduct by product, i.e. disjunction by conjunction, and the
other way round, to obtain the dual results.Comment: 15 page
G\"odel on Deduction
This is an examination, a commentary, of links between some philosophical
views ascribed to G\"odel and general proof theory. In these views deduction is
of central concern not only in predicate logic, but in set theory too,
understood from an infinitistic ideal perspective. It is inquired whether this
centrality of deduction could also be kept in the intensional logic of concepts
whose building G\"odel seems to have taken as the main task of logic for the
future.Comment: 25 page
Ordinals in Frobenius Monads
This paper provides geometrical descriptions of the Frobenius monad freely
generated by a single object. These descriptions are related to results
connecting Frobenius algebras and topological quantum field theories. In these
descriptions, which are based on coherence results for self-adjunctions
(adjunctions where an endofunctor is adjoint to itself), ordinals in epsilon
zero play a prominent role. The paper ends by considering how the notion of
Frobenius algebra induces the collapse of the hierarchy of ordinals in epsilon
zero, and by raising the question of the exact categorial abstraction of the
notion of Frobenius algebra.Comment: 27 page
Bicartesian Coherence
Coherence is demonstrated for categories with binary products and sums, but
without the terminal and the initial object, and without distribution. This
coherence amounts to the existence of a faithful functor from a free category
with binary products and sums to the category of relations on finite ordinals.
This result is obtained with the help of proof-theoretic normalizing
techniques. When the terminal object is present, coherence may still be proved
if of binary sums we keep just their bifunctorial properties. It is found that
with the simplest understanding of coherence this is the best one can hope for
in bicartesian categories. The coherence for categories with binary products
and sums provides an easy decision procedure for equality of arrows. It is also
used to demonstrate that the categories in question are maximal, in the sense
that in any such category that is not a preorder all the equations between
arrows involving only binary products and sums are the same. This shows that
the usual notion of equivalence of proofs in nondistributive
conjunctive-disjunctive logic is optimally defined: further assumptions would
make this notion collapse into triviality. (A proof of coherence for categories
with binary products and sums simpler than that presented in this paper may be
found in Section 9.4 of {\it Proof-Theoretical Coherence}, revised version of
September 2007, http://www.mi.sanu.ac.yu/~kosta/coh.pdf.)Comment: 22 pages, corrected proof of maximalit
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