Riesz Decomposition Property Research Articles

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.

Two-Dimensional Observables and Spectral Resolutions

A two-dimensional observable is a special kind of a σ-homomorphism defined on the Borel σ-algebra of the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra which has properties similar to a two-dimensional distribution function in probability theory. We show that there is a one-to-one correspondence between two-dimensional observables and two-dimensional spectral resolutions defined on a σ-complete MV-algebras as well as on the monotone σ-complete effect algebras with the Riesz decomposition property. The result is applied to the existence of a joint two-dimensional observable of two one-dimensional observables.

Infinite Riesz decompositions

This paper commences the study of infinite analogues of the Riesz decomposition property for ordered vector spaces and ordered normed spaces. A complete result is obtained in the normed case and partial results in general.

Generalized pseudo-EMV-effect algebras

EMV-algebras were recently introduced in Dvurecenskij and Zahiri (Fuzzy Sets Syst, 2019. https://doi.org/10.1016/j.fss.2019.02.013 ) as new structures generalizing both MV-algebras and Boolean rings. These algebras do not assume that they contain a top element. We present a non-commutative generalization of EMV-algebras, called pseudo-EMV-algebras. We show how from a pseudo-EMV-algebra we can derive a generalized pseudo-EMV-effect algebra and conversely, from a generalized effect algebra with a stronger type of the Riesz decomposition property we can derive a pseudo-EMV-algebra. We show that every generalized pseudo-EMV-effect algebra without top element can be embedded into a pseudo-MV-effect algebra with top element as a maximal and normal ideal of the pseudo-MV-effect algebra.

Perfect Effect Algebras and Spectral Resolutions of Observables

We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a $$\hbox {Rad}$$ -monotone $$\sigma $$ -complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral resolutions of observables.

Measures on effect algebras

AbstractWe study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on whichσ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

Logical entropy on effect algebras with the Riesz decomposition property

In this study, the notion of logical entropy is generalized for the case when the considered probability space is an effect algebra with the Riesz decomposition property. We define the logical entropy and conditional logical entropy of finite partitions in an effect algebra with the Riesz decomposition property and prove the basic properties of these measures. Furthermore, we introduce the concepts of logical cross entropy and logical divergence and discuss their desirable properties.

Direct limits of generalized pseudo-effect algebras with the Riesz decomposition properties

In this paper, we focus on direct limits and inverse limits in the category with generalized pseudo-effect algebras (GPEAs for short) as objects and GPEA-morphisms as morphisms. We show that direct limits exist in the category of GPEAs and direct limits of GPEAs satisfy the Riesz decomposition properties whenever the directed systems of GPEAs satisfy the Riesz decomposition properties. Then, we give a condition under which the quotient of a direct limit of GPEAs is a direct limit of quotients of GPEAs. Moreover, we prove that if inverse systems of GPEAs satisfy the Riesz decomposition properties, then inverse limits also satisfy the Riesz decomposition properties.

Order closed ideals in pre-Riesz spaces and their relationship to bands

In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal $I$ in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of $I$ is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal $I$ equals its double disjoint complement, provided the double disjoint complement of $I$ is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples.

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.

When the lexicographic product of two po-groups has the Riesz decomposition property

We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even $${{\rm RDP}_1}$$ , a stronger type of RDP. We recall that a very strong type of RDP, $${{\rm RDP}_2}$$ , entails that the group is lattice ordered. RDP's of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.

Lexicographic pseudo effect algebras

We characterize effect algebras and pseudo effect algebras that can be represented as an interval of the lexicographic product of an antilattice unital po-group (H, u) with a directed po-group G, both with the Riesz Decomposition Property (RDP). We show that a crucial condition is the existence of a lexicographic normal ideal. Finally, we present a categorical equivalence of the category of (H, u)-lexicographic pseudo effect algebras having RDP with the category of directed po-groups with RDP. In addition, a weaker form of the (H, u)-lexicographic pseudo effect algebra is studied.

Kite n-Perfect Pseudo Effect Algebras

Kite pseudo effect algebras were recently introduced as a class of interesting examples of pseudo effect algebras using a po-group, an index set and two bijections on the index set. We represent kite pseudo effect algebras with a special kind of the Riesz decomposition property as an interval in a lexicographic extension of the po-group which solves an open problem on representation of kites. In addition, we introduce kite $n$-perfect pseudo effect algebras and we characterize subdirectly irreducible algebras which are building stones of the theory.

Riesz decomposition properties and the lexicographic product of po-groups

We establish conditions when a certain type of the Riesz decomposition property (RDP) holds in the lexicographic product of two po-groups. It is well known that the resulting product is an $$\ell $$l-group if and only if the first one is linearly ordered and the second one is an $$\ell $$l-group. This can be equivalently studied as po-groups with a special type of the RDP. In the paper we study three different types of RDPs. RDPs of the lexicographic products are important for the study of pseudo effect algebras where infinitesimal elements play an important role both for algebras as well as for the first-order logic of valid but not provable formulas.

Super quantum measures on effect algebras with the Riesz decomposition properties

We give one basis of the space of super quantum measures on finite effect algebras with the Riesz decomposition properties (RDP for short). Then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the RDP are determined each other. At last, we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the RDP in detail.

Unitizations of Generalized Pseudo Effect Algebras and their Ideals

A generalized pseudo effect algebra (GPEA) is a partially ordered partial algebraic structure with a smallest element 0, but not necessarily with a unit (i.e, a largest element). If a GPEA admits a so-called unitizing automorphism, then it can be embedded as an order ideal in its so-called unitization, which does have a unit. We study unitizations of GPEAs with respect to a unitizing automorphism, paying special attention to the behavior of congruences, ideals, and the Riesz decomposition property in this setting.

Quantum Structures Versus Partially Ordered Groups

We present a survey of applications of partially ordered groups (= po-groups) in quantum structures. The survey has two levels: (1) To show when an effect algebra or a pseudo effect algebra is an interval in the positive cone of a po-group. A sufficient condition is a kind of the Riesz Decomposition Property, i.e. a property when two decompositions have a joint refinement. (2) Using an arbitrary power of the positive and negative cone of a po-group and two injections, we can construct a large class of pseudo effect algebras which are called kite pseudo effect algebras, and we describe basic properties of this class.

On a new construction of pseudo effect algebras

We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which are connected not necessarily with partially ordered groups, but rather with generalized pseudo effect algebras where the greatest element is not guaranteed. Starting even with a commutative generalized pseudo effect algebra, we can obtain a non-commutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz decomposition properties. We find conditions when kite pseudo effect algebras have the least non-trivial normal ideal.

Quantum Measures on Finite Effect Algebras with the Riesz Decomposition Properties

One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure $$\lambda $$ on the tensor product $$E\otimes E$$ can determine a quantum measure $$\mu $$ on a finite effect algebra $$E$$ with the RDP such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . Furthermore, some conditions for a grade-2 additive measure $$\mu $$ on a finite effect algebra $$E$$ are provided to guarantee that there exists a unique diagonally positive symmetric signed measure $$\lambda $$ on $$E\otimes E$$ such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . At last, it is showed that any grade- $$t$$ quantum measure on a finite effect algebra $$E$$ with the RDP is essentially established by the values on a subset of $$E$$ .