Lattice Effect Algebras Research Articles

The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states

We study remarkable sub-lattice effect algebras of Archimedean atomic lattice effect algebras E, namely their blocks M, centers C(E), compatibility centers B(E) and sets of all sharp elements S(E) of E. We show that in every such effect algebra E, every atomic block M and the set S(E) are bifull sub-lattice effect algebras of E. Consequently, if E is moreover sharply dominating then every atomic block M is again sharply dominating and the basic decompositions of elements (BDE of x) in E and in M coincide. Thus in the compatibility center B(E) of E, nonzero elements are dominated by central elements and their basic decompositions coincide with those in all atomic blocks and in E. Some further details which may be helpful under answers about the existence and properties of states are shown. Namely, we prove the existence of an (o)-continuous state on every sharply dominating Archimedean atomic lattice effect algebra E with $$B(E)\not =C(E).$$ Moreover, for compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved. Further, we prove “State smearing theorem” for these lattice effect algebras.

Modularity, Atomicity and States in Archimedean Lattice Effect Algebras

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra $E$ that is not an orthomodular lattice there exists an $(o)$-continuous state $\omega$ on $E$, which is subadditive. Moreover, we show properties of finite and compact elements of such lattice effect algebras.

Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra $E$ is separable and modular then there exists a faithful state on $E$. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra $\widehat{E}$ and the compatiblity center of $E$ is not a Boolean algebra then there exists an $(o)$-continuous subadditive state on $E$.

Sharply Orthocomplete Effect Algebras

Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8]. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E. Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete then it is complete.

Effect-like algebras induced by means of basic algebras

Abstract Having an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied.

Topologies on quantum effects

Quantum effects play an important role in quantum measurement theory. The set of all quantum effects can be organized into an algebraical structure called effect algebra. In this paper, we study various topologies on the Hilbert space effect algebra and the projection lattice effect algebra.

A question about basic algebras

We answer in the affirmative a question raised in a recent paper by Chajda, Halas and Kuhr, about characterizing lattice effect algebras among basic algebras.

Common Generalizations of Orthocomplete and Lattice Effect Algebras

Connections between the weak orthocompleteness and the maximality property in effect algebras are presented. It is proved that an orthomodular poset with the maximality property is disjunctive. A characterization of Archimedean weakly orthocomplete effect algebras is given.

Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.

State smearing theorems and the existence of states on some atomic lattice effect algebras

The existence of states and probabilities on effect algebras as logical structures when events may be non-compatible, unsharp, fuzzy or imprecise is still an open question. Only a few families of effect algebras possessing states are known. We are going to show some families of effect algebras, the existence of a pseudocomplementation on which implies the existence of states. Namely, those are Archimedean atomic lattice effect algebras, which are sharply dominating or s-compactly generated or extendable to complete lattice effect algebras.

Interval topology of lattice effect algebras

Let ( E , ⊥ , ⊕ , 0 , 1 ) be a lattice effect algebra and τ i its interval topology. In this work we show that if the operation ⊕ is continuous with respect to τ i , then τ i is Hausdorff. In addition, we also show that if ( E , ⊥ , ⊕ , 0 , 1 ) is a complete M V -effect algebra and τ i is a Hausdorff topology, then the operations ⊕ , ⊖ , ∨ and ∧ are τ i continuous.

Isomorphism theorems on generalized effect algebras based on atoms

A well-known fact is that every generalized effect algebra can be uniquely extended to an effect algebra in which it becomes a sub-generalized effect algebra and simultaneously a proper order ideal, the set-theoretic complement of which is its dual poset. We show that two non-isomorphic generalized effect algebras (even finite ones) may have isomorphic effect algebraic extensions. For Archimedean atomic lattice effect algebras we prove “Isomorphism theorem based on atoms”. As an application we obtain necessary and sufficient conditions for isomorphism of two prelattice Archimedean atomic generalized effect algebras with common (or isomorphic) effect algebraic extensions.

Operation continuity of effect algebras

In this paper, we prove that the operations ⊕ and ⊖ of effect algebras are continuous with respect to their ideal topology, and if the effect algebras are lattice effect algebras, then under some conditions, the lattice operations ∨ and ∧ are also continuous with respect to their ideal topology.

Note on ideals of effect algebras

We show that in a lattice effect algebra, each lattice ideal is an effect algebra ideal if and only if the lattice effect algebra is an orthomodular lattice.

Pseudocomplemented lattice effect algebras and existence of states

We prove that in every pseudocomplemented atomic lattice effect algebra the subset of all pseudocomplements is a Boolean algebra including the set of sharp elements as a subalgebra. As an application, we show families of effect algebras for which the existence of a pseudocomplementation implies the existence of states. These states can be obtained by smearing of states existing on the Boolean algebra of sharp elements.

Frink ideal topology of lattice effect algebras

A difficult question arising in the study of effect algebras is how to equip them with a fitting and proper topology such that the topology is compatible with the partial operations ⊕ and Θ. As we know, the Frink ideal topology is an important intrinsic topology for studying partially ordered set theory; in particular, it is the correct topology for chains and direct products of a finite number of chains. In this paper, we show that the Frink ideal topology is also a nice topology for studying effect-algebra theory since it provides the operations with some of the expected continuity properties.

The Variety of Lattice Effect Algebras Generated by MV-algebras and the Horizontal Sum of Two 3-element Chains

It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum $${\mathcal{H}}$$ of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and $${\mathcal{H}}$$ is definable by three simple identities and the problem now is to check if these identities are satisfied by all distributive lattice effect algebras or not.

States on sharply dominating effect algebras

We prove that Archimedean sharply dominating atomic lattice effect algebras can be characterized by property called basic decomposition of elements. As an application we prove the state smearing theorem for these effect algebras.

Partial residuated structures and quantum structures

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics--classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.

Valuation and metric on a lattice effect algebra

AbstractIn the present paper we have derived some properties of the pseudo-metric introduced by Riečanová on a lattice effect algebra