31,922 research outputs found
Generating finite integral relation algebras
Relation algebras and categories of relations in particular have proven to be extremely useful as a fundamental tool in mathematics and computer science. Since relation algebras are Boolean algebras with some well-behaved operations, every such algebra provides an atom structure, i.e., a relational structure on its set of atoms. In the case of complete and atomic structure (e.g. finite algebras), the original algebra can be recovered from its atom structure by using the complex algebra construction. This gives a representation of relation algebras as the complex algebra of a certain relational structure. This property is of particular interest because storing the atom structure requires less space than the entire algebra. In this thesis I want to introduce and implement three structures representing atom structures of integral heterogeneous relation algebras, i.e., categorical versions of relation algebras. The first structure will simply embed a homogeneous atom structure of a relation algebra into the heterogeneous context. The second structure is obtained by splitting all symmetric idempotent relations. This new algebra is in almost all cases an heterogeneous structure having more objects than the original one. Finally, I will define two different union operations to combine two algebras into a single one
Distributed Relation Logic
We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators.We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras
ReAlM - a system to manipulate relations
Given a heterogeneous relation algebra R, it is well known that the algebra of matrices with coefficient from R is relation algebra with relational sums that is not
necessarily finite. When a relational product exists or the point axiom is given, we
can represent the relation algebra by concrete binary relations between sets, which
means the algebra may be seen as an algebra of Boolean matrices. However, it is
not possible to represent every relation algebra. It is well known that the smallest
relation algebra that is not representable has only 16 elements. Such an algebra can
not be put in a Boolean matrix form.[15]
In [15, 16] it was shown that every relation algebra R with relational sums and
sub-objects is equivalent to an algebra of matrices over a suitable basis. This basis is
given by the integral objects of R, and is, compared to R, much smaller.
Aim of my thesis is to develop a system called ReAlM - Relation Algebra Manipulator - that is capable of visualizing computations in arbitrary relation algebras
using the matrix approach
Non normal logics: semantic analysis and proof theory
We introduce proper display calculi for basic monotonic modal logic,the
conditional logic CK and a number of their axiomatic extensions. These calculi
are sound, complete, conservative and enjoy cut elimination and subformula
property. Our proposal applies the multi-type methodology in the design of
display calculi, starting from a semantic analysis based on the translation
from monotonic modal logic to normal bi-modal logic
Formalization of Universal Algebra in Agda
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin
Varieties whose tolerances are homomorphic images of their congruences
The homomorphic image of a congruence is always a tolerance (relation) but,
within a given variety, a tolerance is not necessarily obtained this way. By a
Maltsev-like condition, we characterize varieties whose tolerances are
homomorphic images of their congruences (TImC). As corollaries, we prove that
the variety of semilattices, all varieties of lattices, and all varieties of
unary algebras have TImC. We show that a congruence n-permutable variety has
TImC if and only if it is congruence permutable, and construct an idempotent
variety with a majority term that fails TImC
- …