Pseudo Effect Algebras Research Articles

Non-commutative effect algebras, L-algebras, and local duality

Abstract GPE-algebras were introduced by Dvurečenskij and Vetterlein as unbounded pseudo-effect algebras. Recently, they have been characterized as partial L-algebras with local duality. In the present paper, GPE-algebras with an everywhere defined L-algebra operation are investigated. For example, linearly ordered GPE-algebra are of that type. They are characterized by their self-similar closures which are represented as negative cones of totally ordered groups. More generally, GPE-algebras with an everywhere defined multiplication are identified as negative cones of directed groups. If their partial L-algebra structure is globally defined, the enveloping group is lattice-ordered. For any self-similar L-algebra A, exponent maps are introduced, generalizing conjugation in the structure group. It is proved that the exponent maps are L-algebra automorphisms of A if and only if A is a GPE-algebra. As an application, a new characterization of cone algebras is obtained. Lattice GPE-algebras are shown to be equivalent to ∧-closed L-algebras with local duality.

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.

Q-residuated lattices and lattice pseudoeffect algebras

Q-residuated lattices and lattice pseudoeffect algebras

Residuation in lattice effect algebras

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed from its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced by Dvurečenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.

Uniqueness of algebraic unitizations of generalized pseudo effect algebras

We study the uniqueness of the algebraic unitizations of a generalized pseudo effect algebra (GPEA). We prove that for a weakly commutative GPEA, if its unitization is a weakly PEA, then it is unique in the sense of PEA-isomorphism.

Pseudo effect algebras are algebras over bounded posets

We prove that there is a monadic adjunction between the category of bounded posets and the category of pseudo effect algebras.

A semiring-like representation of lattice pseudoeffect algebras

In order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized ?ukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations.

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.

Congruences and ideals in generalized pseudoeffect algebras revisited

The first part of the present paper is an enhancement of the paper (Foulis et al. in Order 33:311–332, 2016). A new type of congruences on generalized pseudoeffect algebras (GPEAs), called R1-congruences, is introduced, which is in one-to-one correspondence with normal R1-ideals. The notion of Riesz congruences is reconsidered, and they are defined as congruences which are in one-to-one correspondence with normal Riesz ideals. In upward directed GPEAs, in particular in pseudoeffect algebras, both these types of congruences as well as ideals coincide. Conditions under which congruences and ideals in a GPEA P may be extended to a \(\gamma \)-unitization of U of P are clarified. In the last part of the paper, subcentral and central ideals in GPEAs and their relations to subdirect and direct decompositions are studied.

Measure games on pseudo-D-lattices

Abstract We prove linearity theorems for modular measures on pseudo-D-lattices (= lattice ordered pseudo-effect algebras) and we study the consequences for the core of measure games.

Uniqueness theorems for finitely additive probabilities on quantum structures

Abstract The proofs of uniqueness theorems, presented here, allow to extend the earlier results. For example, the following hold: let μ and ν be two finitely additive probabilities on a structure L̃ (for example, L̃ is a pseudo-effect algebra), and let μ be convex-ranged; if there exists an element a ∈ L̃ with 0 < μ(a) < 1 and such that μ(a) = μ(b) ⇒ ν(a) = ν(b) whenever b ∈ L̃, then ν = μ.

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.

Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem

We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every \(\mathbb {R}^{n} \)-valued σ-additive modular measure on a σ-complete pseudo-D-lattice.

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.

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.

Pseudo-D-lattices and separating points of measures

In 1986, A. Basile and H. Weber proved that every countable family M of σ-additive measures on a σ-complete Boolean ring R admits a dense Gδ-set of separating points, with respect to a suitable topology, in the following two cases: either (i) each element of M is s-bounded and, whenever λ,ν∈M are distinct, the quotient of R modulo N(λ−ν) is infinite, or (ii) each element of M is continuous. In this paper we consider modular measures on lattice-ordered pseudo-effect algebras. Using topological methods, we extend the result of Basile and Weber in a way that allows to unify the above two cases (i) and (ii). This gives a new contribution also in the classical setting of algebras of sets.

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.

Prelattice-generalized pseudo-effect algebras

Using the unitizations of symmetric-generalized pseudo-effect algebras, we investigate the relationships between the maximal compatible sets in a prelattice symmetric-generalized pseudo-effect algebra and the maximal compatible sets in its unitization. We also prove that a symmetric-generalized pseudo-effect algebra is a symmetric-generalized pseudo MV-effect algebra if and only if its unitization is a symmetric pseudo MV-effect algebra.