Effect Algebras Research Articles

A Categorical Construction of the Real Unit Interval

The real unit interval is the fundamental building block for many branches of mathematics like probability theory, measure theory, convex sets and homotopy theory. However, a priori the unit interval could be considered an arbitrary choice and one can wonder if there is some more canonical way in which the unit interval can be constructed. In this paper we find such a construction by using the theory of effect algebras. We show that the real unit interval is the unique non-initial, non-final irreducible algebra of a particular monad on the category of bounded posets. The algebras of this monad carry an order, multiplication, addition and complement, and as such model much of the operations we need to do on probabilities. On a technical level, we show that both the categories of omega-complete effect algebras as well as that of omega-complete effect monoids are monadic over the category of bounded posets using Beck's monadicity theorem. The characterisation of the real unit interval then follows easily using a recent representation theorem for omega-complete effect monoids.

The Range of a Module Measure Defined on an Effect Algebra

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algebra. The range of a module measure μ:L→M is studied. Among other results, we prove that if L is an sRDP σ-effect algebra with a natural basis and μ:L→R is a countably additive measure, then μ has bounded variation.

Representation of perfect and n-perfect pseudo effect algebras

A perfect (an n-perfect) pseudo effect algebra can be decomposed into two (n+1 many) non-empty and mutually comparable slices. They generalize perfect MV-algebras studied in [5]. We characterize such a pseudo effect algebra as an interval in the semidirect product of the po-group Z or 1nZ with a directed po-group G satisfying a stronger type of the Riesz Decomposition Property, RDP1, and the semidirect product is ordered lexicographically. We show that the category of perfect and the category of n-perfect pseudo effect algebras with RDP1 are categorically equivalent to a special category of directed po-groups satisfying RDP1.

Homogeneous Effect Algebras and Observables vs Spectral Resolutions

Homogeneous Effect Algebras and Observables vs Spectral Resolutions

Regular Partial Residuated Lattices and Their Filters

To express wider uncertainty, Běhounek and Daňková studied fuzzy partial logic and partial function. At the same time, Borzooei generalized t-norms and put forward the concept of partial t-norms when studying lattice valued quantum effect algebras. Based on partial t-norms, Zhang et al. studied partial residuated implications (PRIs) and proposed the concept of partial residuated lattices (PRLs). In this paper, we mainly study the related algebraic structure of fuzzy partial logic. First, we provide the definitions of regular partial t-norms and regular partial residuated implication (rPRI) through the general forms of partial t-norms and partial residuated implication. Second, we define regular partial residuated lattices (rPRLs) and study their corresponding properties. Third, we study the relations among commutative quasi-residuated lattices, commutative Q-residuated lattices, partial residuated lattices, and regular partial residuated lattices. Last, in order to obtain the filter theory of regular partial residuated lattices, we restrict certain conditions and then propose special regular partial residuated lattices (srPRLs) in order to finally construct the quotient structure of regular partial residuated lattices.

Convexity-preserving Properties of Partial Binary Operations with Respect to Filter Convex Structures on Effect Algebras

Convexity-preserving Properties of Partial Binary Operations with Respect to Filter Convex Structures on Effect Algebras

New results on topological effect algebras

In this paper, by considering the notion of effect algebra and by using of a new ideal in an effect algebra E, we construct a topology τ on E, and we show that (E,τ) is a topological effect algebra. Then we obtain some conditions under which that (E,τ) is a Hausdorff space. Also, we obtain some results about connected components of this topological space, and we construct a quotient topological effect algebra.

Observables on σ-Frame effect Algebras as Upper Semicontinuous Functions

We showed that the set of observables O(E) on a σ-frame effect algebra E is in one-to-one correspondence with the set USC(is) of point-free versions of real upper semicontinous functions on the order reduct of E. The Olson order and the sum on O(E) are represented by the usual order and the sum of real upper semicontinuous functions. Sharp observables SO(E) are then described by continuous functions on E and this correspondence is one-to-one in the case when E forms an MV-algebra.

Results on Topological Lattice Effect Algebras

In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a $T_{0}$, $T_{1}$, and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.

The effective Dirac algebra by gauge field interaction in relativistic electrodynamics

Conventional relativistic electrodynamics is set on flat Minkowski spacetime, where all computable quantities are calculated from the flat metric [Formula: see text]. We can redefine the metric of spacetime from the Dirac algebra. In this paper, we study how an electrodynamic interaction can alter the normal gamma matrix to an effective one and result in a shift in the metric perturbatively. The curvature properties inferred from the curved metric are also investigated. We also study how the spin operator is changed under the interaction that contributes to an effective spin operator and how the spin of an electron will be slightly deviated from [Formula: see text]. Then we perform canonical quantization of the effective Dirac algebra. Finally, we apply our results to the relativistic hydrogen case and demonstrate how such system curves the spacetime metric.

Research on Partial Ordered Sets That Can Be Constructed as Effect Algebras

In this paper, we study effect algebra‐induced partial ordered sets. All possible cases of effect algebras generated by bounded partial ordered set MI of height 2 are given. In addition, the structure of chain effect algebra is studied carefully and the corresponding results are obtained.

Monadic Effect Algebras

The main goal of this paper is to introduce and investigate the related theory on monadic effect algebras. First, we design the axiomatic system of existential quantifiers on effect algebras and then use it to give the definition of the universal quantifier and monadic effect algebras. Then, we introduce relatively complete subalgebra and prove that there exists a one‐to‐one correspondence between the set of all the existential quantifiers and the set of all the relatively complete subalgebras. Moreover, we characterize and give the generated formula of monadic ideals and prove that Riesz monadic ideals and Riesz monadic congruences can be mutually induced. Finally, we study the strong existential quantifier and characterize monadic simple and monadic subdirectly irreducible effect algebras.

On the Entropy of Partitions in Hyper Effect Algebra

On the Entropy of Partitions in Hyper Effect Algebra

On the effective universality of mereological theories

AbstractMereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first‐order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first‐order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important computability‐theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory T, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory T is ‐universal. We obtain the following results about known mereological theories. Any theory T which is weaker than Extensional Closure Mereology (CEM) is ‐universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not ‐universal. In particular, General Extensional Mereology is not ‐universal.

N-dimensional observables on k-perfect MV-algebras and k-perfect effect algebras. II. One-to-one correspondence

This article is a continuation of our research on a one-to-one correspondence between n-dimensional spectral resolutions and n-dimensional observables on lexicographic types of quantum structures which started in Dvurečenskij and Lachman (https://doi.org/10.1016/j.fss.2021.05.005). There we presented the main properties of n-dimensional spectral resolutions and observables, and we studied in depth characteristic points which are crucial for our study. Here we present the main body of our research. We investigate a one-to-one correspondence between n-dimensional observables and n-dimensional spectral resolutions with values in a lexicographic form of quantum structures such as perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is k-perfect for k>1, then even for the two-dimensional case of spectral resolutions we have more characteristic points. The results obtained are applied to the existence of an n-dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra and a sum of n-dimensional observables.

Residuation in finite posets

Abstract When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

L-Fuzzy Sub-Effect Algebras

In this paper, the notions of L-fuzzy subalgebra degree and L-subalgebras on an effect algebra are introduced and some characterizations are given. We use four kinds of cut sets of L-subsets to characterize the L-fuzzy subalgebra degree. We induce an L-fuzzy convexity by the L-fuzzy subalgebra degree, and we prove that a morphism between two effect algebras is an L-fuzzy convexity preserving mapping and a monomorphism is an L-fuzzy convex-to-convex mapping. Finally, it is proved that the set of all L-subalgebras on an effect algebra can form an L-convexity, and its L-convex hull formula is given.

Properties of implication in effect algebras

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

Closure systems over effect algebras

The present paper is an attempt to introduce the closure systems over effect algebras. At first, we will define closure systems over effect algebras, and for arbitrary set $ U $ and arbitrary subset S of all functions from U to an effect algebra L we will obtain the closure system containing S. Then, we will define the base of this closure system, and for arbitrary subset S of all functions from U to an effect algebra L we will obtain the base of this closure system.

N-dimensional observables on k-perfect MV-algebras and k-perfect effect algebras. I. Characteristic points

In the paper, we investigate a one-to-one correspondence between n-dimensional observables and n-dimensional spectral resolutions with values in a kind of a lexicographic form of quantum structures like perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is k-perfect for k>1, then even for the two-dimensional case we have more characteristic points. The obtained results are also applied to existence of an n-dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra. The results are divided into two parts. In Part I, we present notions of n-dimensional observables and n-dimensional spectral resolutions with accent on lexicographic type effect algebras and lexicographic MV-algebras. We concentrate on characteristic points of spectral resolutions and the main body is in Part II where one-to-one relations between observables and spectral resolutions are presented.