Lattice Effect Algebras Research Articles

Interval convexity of scale effect algebras

In this paper, we first propose the concept of DC-nets, where its domain is a DCPO, in convexity spaces and discuss some related properties. And then, we obtain that is convexity-preserving if and only if converges to f(a) with respect to when converges to a with respect to Hence, we could discuss convexity-preserving properties of partial binary operations + and – of effect algebras by convergence of DC-nets in convexity spaces. Concretely, we prove that if is a scale effect algebra and is an interval convexity on E, then + and – are separately convexity-preserving with respect to . Finally, we provide an example to show that + and – are not jointly convexity-preserving with respect to when is a lattice effect algebra. Communicated by Ángel del Río Mateos

Partial Residuated Implications Induced by Partial Triangular Norms and Partial Residuated Lattices

This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce the connection between lattice effect algebra and partial t-norms according to the concept of partial t-norms given by Borzooei, together with the proof that partial operation in any commutative quasiresiduated lattice is partial t-norm. Then, we offer the general form of PRI and the definition of partial fuzzy implication (PFI), give the condition that partial residuated implication is a fuzzy implication, and prove that each PRI is a PFI. Next, we propose PRLs, study their basic characteristics, discuss the correspondence between PRLs and lattice effect algebras (LEAs), and point out the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we provide the notions of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Lastly, based on partial residuated lattices, we define well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of PRMs.

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 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.

Fuzzy implications in lattice effect algebras

Implications are one of the key operations in fuzzy logic and approximate reasoning. We prove that the Dishkant arrow in a lattice effect algebra is a partial triangular implication if and only if the lattice effect algebra is an MV-effect algebra. Then we show that a certain implication based on the Sasaki arrow is a partial triangular implication. Furthermore, we propose a new partial triangular implication, which is also a fuzzy implication in lattice effect algebras.

The structure group of a non-degenerate effect algebra

A (non-commutative) structure group G(E) is associated to an arbitrary effect algebra E, and a concept of non-degeneracy is introduced. If E is non-degenerate, G(E) has a right invariant partial order, E embeds as an interval into G(E), and the negative cone of G(E) is a self-similar partial L-algebra. In the degenerate case, the possible anomalies are explained. Lattice effect algebras, 2-divisible effect algebras, and the effect algebra of a Hilbert space, are shown to be non-degenerate. As an application, using a strengthened concept of sharpness, the theory of block decomposition is extended to arbitrary effect algebras, and it is shown that the “very sharp” elements always form a sub-effect algebra and an orthomodular poset.

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.

Quotient structures in lattice effect algebras

Quotient structures in lattice effect 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.

L-effect Algebras

L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a $$\wedge $$ -semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S(X) form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right $$\ell $$ -group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.

A Complete Axiomatisation for the Logic of Lattice Effect Algebras

In a recent work Foulis and Pulmannov\' a \cite{Foulis2012} studied the logical connectives in lattice effect algebras. In this paper we extend their study and investigate further the logical calculus for which the lattice effect algebras can serve as semantic models. We shall first focus on some properties of lattice effect algebras and will then give a complete axiomatisation of this logic.

EMV-pairs

ABSTRACTThe class of EMV-algebras and the class of lattice generalized effect algebras with RDP which have enough idempotents (EMV-effect algebra for short) are termwise equivalent. We use it to generalize connections between MV-algebras and MV-pairs for the case of EMV-algebras. First, we show that each EMV-algebra M is a homomorphic image of , the generalized Boolean algebra R-generated by M. Then we introduce a concept of an EMV-pair and construct an EMV-algebra using the EMV-pair. We also show that each EMV-algebra can be induced in this way. Then, the concept of an EMV-pair as a generalization of an MV-pair is introduced and some related results are obtained. Finally, we study G-invariant ideals of an EMV-pair .

Fuzzy rough set based on lattice effect algebra

Fuzzy rough set based on lattice effect algebra

The sum of observables on a $$\sigma $$ σ -distributive lattice effect algebra

Observables on quantum structures can be seen as generalizations of random variables on a measurable space $$(\Omega , \mathcal {A})$$ for the case when $$\mathcal {A}$$ is not necessarily a Boolean algebra. The present paper investigates an extending of the usual pointwise sum of random variables onto the set of bounded observables on a $$\sigma $$ -distributive lattice effect algebra E. We describe conditions under which this operation, so-called sum $$x+y$$ of observables x, y, preserves continuity of spectral resolutions of x, y. We show how the spectrum $$\sigma (x+y)$$ depends on spectra $$\sigma (x)$$ , $$\sigma (y)$$ , and we provide a relation between the meager part $$x_m$$ and the dense part $$x_d$$ of an observable x.

Hyper-commutators in effect algebras

The hyper-commutators in effect algebras as a generalization of commutators in lattice effect algebras are defined, and their properties are studied. We prove that a nonzero element is in some hyper-commutator if and only if it is a torsion element in effect algebras with the maximality property. Thus, homogeneity can be characterized by hyper-commutators. Some properties of Riesz ideals related to hyper-commutators are also studied. As an application, we prove that commutators are sharp in lattice effect algebras and every block-finite lattice effect algebra is commutator-finite.

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.

Material implications in lattice effect algebras

In this paper we show that it is not acceptable using the minimal conditions of implication introduced for orthomodular lattices as minimal conditions of a good implication for lattice effect algebras. Hence, we first define the concept of a partial t-norm on bounded lattices. Then we obtain the concept of a pt-implication on bounded involutive lattices by introducing some conditions using partial t-norms. Furthermore, we conclude that the Sasaki arrow which is a very important implication in lattice effect algebras is the best pt-implication on them. Finally, applying the Sasaki arrow, we construct a fuzzy implication on a lattice effect algebra and also we prove that it is a weak pt-implication as well.

Meager projections in orthocomplete homogeneous effect algebras

We define meager projections on homogeneous effect algebras and discuss their properties. As an application, we prove that, if E is an orthocomplete homogeneous effect algebra such that S(E) is lattice-ordered, then E is a lattice effect algebra, which gives an affirmative answer to an open problem stated by Mesiar and Stupňanová.

A triple representation of lattice effect algebras

Abstract A mutual relationship between MV-algebras and coupled semirings as established by L. P. Belluce, A. Di Nola, A. R. Ferraioli and B. Gerla is extended to lattice effect algebras and so-called characterizing triples. We show that this correspondence is in fact one-to-one and hence every lattice effect algebra can be considered as an ordered triple consisting of two semiring-like structures and an antitone involution which is an isomorphism between these structures.

A Representation of Lattice Effect Algebras by Means of Right Near Semirings with Involution

Since every lattice effect algebra decomposes into blocks which are MV-algebras and since every MV-algebra can be represented by a certain semiring with an antitone involution as shown by Belluce, Di Nola and Ferraioli, the natural question arises if a lattice effect algebra can also be represented by means of a semiring-like structure. This question is answered in the present paper by establishing a one-to-one correspondence between lattice effect algebras and certain right near semirings with an antitone involution.