204 research outputs found
The recognition problem for table algebras and reality-based algebras
Given a finite-dimensional noncommutative semisimple algebra with
involution, we show that always has an RBA-basis. We look for an RBA-basis
that has integral or rational structure constants, and ask if the RBA admits a
positive degree map. For RBAs that have a positive degree map, we try to find
an RBA-basis with nonnegative structure constants to determine if there is a
generalized table algebra structure. We settle these questions for the algebras
, .Comment: 16 page
Morphisms and Duality for Polarities and Lattices with Operators
Structures based on polarities have been used to provide relational semantics
for propositional logics that are modelled algebraically by non-distributive
lattices with additional operators. This article develops a first order notion
of morphism between polarity-based structures that generalises the theory of
bounded morphisms for Boolean modal logics. It defines a category of such
structures that is contravariantly dual to a given category of lattice-based
algebras whose additional operations preserve either finite joins or finite
meets. Two different versions of the Goldblatt-Thomason theorem are derived in
this 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
Low-Dimensional Reality-Based Algebras
In this paper we introduce the definition of a reality-based algebra (RBA) as well as a subclass of reality-based algebras, table algebras. Using sesquilinear forms, we prove that a reality-based algebra is semisimple. We look at a specific reality-based algebra of dimension 5 and provide formulas for the structure constants of this algebra. We determine by looking at these structure constants and setting conditions on specific structural components when this particular reality-based algebra is a table algebra. In fact, this will be a noncommutative table algebra of dimension 5
- β¦