Lattice-ordered Effect Algebras Research Articles

Basic algebras and L-algebras

In this paper, we study the relation between L-algebras and basic algebras. In particular, we construct a lattice-ordered effect algebra which improves an example of Chajda et al. (Algebra Univ 60(1), 63–90, 2009).

The logic induced by effect algebras

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras {mathbf {E}}, we investigate a natural implication and prove that the implication reduct of {mathbf {E}} is term equivalent to {mathbf {E}}. Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

LYAPUNOV DECOMPOSITION OF MEASURES ON EFFECT ALGEBRAS

We prove that every closed exhaustive vector-valued modular measure on a lattice ordered effect algebra L can be decomposed into the sum of a Lyapunov exhaustive modular measure (i.e. its restriction to every interval of L has convex range) and an ”antiLyapunov” exhaustive modular measure. This result extends a Kluvanek-Knowles decomposition theorem for measures on Boolean algebras.

Lattice-ordered effect algebras and L-algebras

It is known that many logical algebras can be considered as L-algebras. In this paper, we first construct an effect algebra (E,⊕,0,1) that cannot be converted into an L-algebra (E,→,0,1) with negation such that x→0 is the orthosupplement. Then we prove that every lattice-ordered effect algebra gives rise to an L-algebra with the same orthosupplement. Finally, we characterize LE-L-algebras as lattice-ordered effect algebras. As an application, we extend those results obtained in some recent references for orthomodular lattices and MV-algebras.

Outer measure on effect algebras

Abstract In the present paper, the notion of an outer measure m * on lattice ordered effect algebras L is introduced and investigated. Carathéodory’s criterion for the outer measure m * is given and established on L. Properties of an induced outer measure m * defined on a lattice ordered effect algebra L are studied and finally, Choquet theorem is proved for the induced outer measure m * on a σ-complete lattice ordered effect algebra L.

The pasting constructions for effect algebras

Abstract The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

Pasting of lattice-ordered effect algebras

In this study, we present techniques for constructing a lattice-ordered effect algebra using a given family of MV-effect algebras. First, we introduce the Greechie diagrams of the pastings of a family of MV-effect algebras. As application of the Greechie diagrams, we provide some sufficient conditions for pasting a lattice-ordered effect algebra using a family of MV-effect algebras. Especially, we illustrate a kind of technique of how to obtain a lattice-ordered effect algebra by substituting the atoms of an orthomodular lattice with the lattice-ordered effect algebras. Finally, we prove a version of the “Loop Lemma” for MV-effect algebras pasting.

Roughness in lattice ordered effect algebras.

Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras.

A Note of Filters in Effect Algebras

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras

Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.

Finitely generated varieties of distributive effect algebras

Lattice-ordered effect algebras generalize both MV-algebras and orthomodular lattices. In this paper, finitely generated varieties of distributive lattice effect algebras are axiomatized, and for any positive integer n, the free n-generator algebras in these varieties are described.

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ℓ-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ℓ-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ℓ-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.

A theory of computation based on unsharp quantum logic: Finite state automata and pushdown automata

When generalizing the projection-valued measurements to the positive operator-valued measurements, the notion of the quantum logic generalizes from the sharp quantum logic to the unsharp quantum logic. It is known that: (i) the distributive law is one of the main differences between the sharp quantum logic and the boolean logic, and the block or the center of the sharp quantum structures are boolean algebras; (ii) the unsharp quantum logic does not satisfy the non-contradiction law, which forces the block or the center of unsharp quantum structures to be multiple valued algebras, rather than boolean algebras. Multiple valued algebras, as special quantum structures, are the algebraic semantics of multiple valued logic. Interestingly, we recently discovered that the difference between some unsharp quantum structures and multiple valued algebras is also some kind of distributive law.Choosing an orthomodular lattice (an algebraic model of a sharp quantum logic) to be the truth valued lattice, Ying et al. have systematically developed automata theory based on sharp quantum logic. In this paper, choosing a lattice ordered quantum multiple valued algebra E (an extended lattice ordered effect algebra E, respectively) to be the truth valued lattice, we also systematically develop an automata theory based on unsharp quantum logic. We introduce E-valued finite-state automata and E-valued pushdown automata in the framework of unsharp quantum logic. We study the classes of languages accepted by these automata and re-examine their various properties in the framework of unsharp quantum logic. The study includes the equivalence between finite-state automata and regular expressions, as well as the equivalence between pushdown automata and context-free grammars. It is also demonstrated that the universal validity of some important properties (such as some closure properties of languages and Kleene theorem etc.) depends heavily on the aforementioned distributive law. More precisely, when the underlying model degenerates into an MV algebra, then all the counterparts of properties in classical automata are valid. This is the main difference between automata theory based on unsharp quantum logic and automata theory based on sharp quantum logic.

The pasting constructions of lattice ordered effect algebras

The aim of this paper is to present some ways of constructing a lattice ordered effect algebra from the given family of MV-algebras. The loop lemma of lattice ordered effect algebras is proved. Then the definitions of Greechie diagrams of effect algebras are given. All these results have generalized those of orthomodular lattices. As applications of loop lemma and Greechie diagrams, some lattice ordered effect algebras without states are presented.

Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x↓,x] is a subset of B. For every meager element (that means, an element x with x↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2M(E) given by h(a) = [0,a] ∩ M(E).

An extension theorem for modular measures on effect algebras

We prove an extension theorem for modular measures on lattice ordered effect algebras. This is used to obtain a representation of these measures by the classical ones. With the aid of this theorem we transfer control theorems, Vitali-Hahn-Saks, Nikodým theorems and range theorems to this setting.

Many-valued quantum algebras

We deal with algebras \({\bf A} = (A, \oplus, \neg, 0)\) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which \(\neg a\) is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.

Decomposition of effect algebras and the Hammer–Sobczyk theorem

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer�Sobczyk type decomposition theorem for modular measures on D-lattices.

On the Aumann–Shapley value

Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley value φ on \(\mathfrak{bv}^\prime\)NA, the subspace of BV spanned by all functions of the form \(f\circ\mu\), where \(\mu: L \to [0,1]\) is a non-atomic σ-additive modular measure and \(f: [0,1] \to {\mathbb{R}}\) is of bounded variation and continuous at 0 and at 1.

The block structure of complete lattice ordered effect algebras

AbstractWe prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).