Universal Associative Geometry

We generalize parts of the theory of associative geometries developed by Kinyon and the author in the framework of universal algebra: we prove that certain associoid structures, such as pregroupoids and principal equivalence relations, have a natural prolongation from a set to its the power set. We reinvestigate the case of homogeneous pregroupoids (corresponding to the projective geometry of a group) from the point of view of pairs of commuting principal equivalence relations. We use the ternary approach to groupoids developed by Anders Kock, and the torsors defined by our construction can be seen as a generalisation of the known groups of bisections of a groupoid.Comment: several figures ; typeset with XeLate

Generalised bialgebras and entwined monads and comonads

Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincar\'e-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework.Comment: 17 page

A Bridge Between Q-Worlds

Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `Q-worlds'. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.Comment: v2: 40 pages, latex. Typos are corrected and 4 references are added. Readability of some proofs are improved with intermediate steps in the formulae. Discussions on weakly self-adjoint operators are moved to the forthcoming pape

Can the game be quantum?

The game in which acts of participants don't have an adequate description in terms of Boolean logic and classical theory of probabilities is considered. The model of the game interaction is constructed on the basis of a non-distributive orthocomplemented lattice. Mixed strategies of the participants are calculated by the use of probability amplitudes according to the rules of quantum mechanics. A scheme of quantization of the payoff function is proposed and an algorithm for the search of Nash equilibrium is given. It is shown that differently from the classical case in the quantum situation a discrete set of equilibrium is possible.Comment: LATEX, 26 pages, 8 figure

Toward a More Natural Expression of Quantum Logic with Boolean Fractions

The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or conditional propositions). This system of ordered pairs (A|B) of events A, B, can express all of the non-Boolean aspects of quantum logic without having to resort to a more abstract formulation like Hilbert space. Such notions as orthogonality, superposition, simultaneous verifiability, compatibility, orthoalgebras, orthocomplementation, modularity, and the Sasaki projection mapping are translated into this conditional event framework and their forms exhibited. These concepts turn out to be quite adequately expressed in this near-Boolean framework thereby allowing more natural, intuitive interpretations of quantum phenomena. Results include showing that two conditional propositions are simultaneously verifiable just in case the truth of one implies the applicability of the other. Another theorem shows that two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b=d, that their conditions are equivalent. Some concepts equivalent in standard formulations of quantum logic are distinguishable in the conditional event algebra, indicating the greater richness of expression possible with Boolean fractions. Logical operations and deductions in the linear subspace logic of quantum mechanics are compared with their counterparts in the conditional event realm. Disjunctions and implications in the quantum realm seem to correspond in the domain of Boolean fractions to previously identified implications with respect to various naturally arising deductive relations.Comment: 46 pages, 2 figures for submittal to a refereed journa

A Categorical Outlook on Cellular Automata

In programming language semantics, it has proved to be fruitful to analyze context-dependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors of cellular automata with coKleisli maps of the exponent comonad on the category of uniform spaces and uniformly continuous functions and exploit this equivalence to conclude some standard results about cellular automata as instances of basic category-theoretic generalities. In particular, we recover Ceccherini-Silberstein and Coornaert's version of the Curtis-Hedlund theorem.Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010

Flat Coset Decompositions of Skew Lattices

Skew lattices are non-commutative generalizations of lattices, and the cosets represent the building blocks that skew lattices are built of. As by Leech's Second Decomposition Theorem any skew lattice embeds into a direct product of a left-handed skew lattice by a right-handed one, it is natural to consider the so called flat coset decompositions, i.e. decompositions of a skew lattice into right and left cosets, thus finding the smallest atoms that compose the structure.Comment: 26 page

Quantum Markov Networks and Commuting Hamiltonians

Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum many-body systems---complementing the area law at finite temperature---and are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more general graphs we demonstrate the equivalence between quantum Markov networks and Gibbs states of a class of Hamiltonians of intermediate complexity between commuting and general local Hamiltonians

Ultrafilters, finite coproducts and locally connected classifying toposes

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin--Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC(Set,Set) of finite-coproduct-preserving endofunctors of Set is equivalent to the presheaf category [UF,Set]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n-types and n-tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories. As a further application of our main result, we use it to describe a first-order analogue of J\'onsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum--Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete.Comment: 32 pages; v2: final journal versio

Superlogic Manifolds and Geometric approach to Quantum Logic

The main purpose of this paper is to present a new approach to logic or what we will call superlogic. This approach constitutes a new way of looking at the connection between quantum mechanics and logic. It is a {\it geometrisation} of the quantum logic. Note that this superlogic is not distributive reflecting a good propriety to describe quantum mechanics, non commutative spaces and contains a nilpotent element.Comment: 10 page