Related papers: Scaled Boolean Algebras
A Boolean $\sigma$-algebra $B$ is a measure algebra if and only if it is weakly distributive and uniformly concentrated.
Finding solution values for unknowns in Boolean equations was a principal reasoning mode in the Algebra of Logic of the 19th century. Schr\"oder investigated it as Aufl\"osungsproblem (solution problem). It is closely related to the modern…
The operations of linear algebra, calculus, and statistics are routinely applied to measurement scales but certain mathematical conditions must be satisfied in order for these operations to be applicable. We call attention to the conditions…
A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
Clones are specializations of operads forming powerful instruments to describe varieties of algebras wherein repeating variables are allowed in their equations. They allow us in this way to realize and study a large range of algebraic…
In this paper, we study nonlinear differential equations satisfied by the generating function of Boole numbers. In addition, we derive some explicit and new interesting identities involving Boole numbers and higher-order numbers arising…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
In statistics cumulants are defined to be functions that measure the linear independence of random variables. In the non-communicative case the Boolean cumulants can be described as functions that measure deviation of a map between algebras…
Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…
This is a proposal of an algebra which aims at distributed array processing. The focus lies on re-arranging and distributing array data, which may be multi-dimensional. The context of the work is scientific processing; thus, the core…
Calibration is a classical notion from the forecasting literature which aims to address the question: how should predicted probabilities be interpreted? In a world where we only get to observe (discrete) outcomes, how should we evaluate a…
Scaling arguments provide valuable analysis tools across physics and complex systems yet are often employed as one generic method, without explicit reference to the various mathematical concepts underlying them. A careful understanding of…
P-algebras are a non-commutative, non-associative generalization of Boolean algebras that are for quantum logic what Boolean algebras are for classical logic. P-algebras have type <X, 0, ', .> where 0 is a constant, ' is unary and . is…
Curved Boolean Logic (CBL) generalizes propositional logic by allowing local truth assignments that do not extend to a single global valuation, analogous to curvature in geometry. We give equivalent sheaf and exclusivity-graph semantics and…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
This paper studies the important problem of quantum classification of Boolean functions from a entirely novel perspective. Typically, quantum classification algorithms allow us to classify functions with a probability of $1.0$, if we are…
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1…