Riesz Decomposition Property Research Articles

Every state on interval effect algebra is integral

We show that every state on an interval effect algebra is an integral through some regular Borel probability measure defined on the Borel σ-algebra of a compact Hausdorff simplex. This is true for every effect algebra satisfying Riesz decomposition property or for every many valued (MV)-algebra. In addition, we show that each state on an effect subalgebra of an interval effect algebra E can be extended to a state on E. Our method represents also every state on the set of effect operators of a Hilbert space as an integral.

On two versions of the Loomis–Sikorski Theorem for algebraic structures

We present two versions of the Loomis---Sikorski Theorem, one for monotone ?-complete generalized pseudo effect algebras with strong unit satisfying a kind of the Riesz decomposition property. The second one is for Dedekind ?-complete positive pseudo Vitali spaces with strong unit. For any case we can find an appropriate system of nonnegative bounded functions forming an algebra of the given type with the operations defined by points that maps epimorphically onto the algebra.

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples.

Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

AbstractWe introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP.

An isomorphic characterization of L1-spaces

We show that a sequentially (τ)-complete topological vector lattice X τ is isomorphic to some L 1(μ), if and only if the positive cone can be written as X + = ℝ + B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X + \\ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness ( monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for ( AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y + = ℝ + Y + = ℝ + B ¯ ( u , 1 ) (for ‖ u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.

Residuated lattices and lattice effect algebras

Residuated lattices and lattice effect algebras arose in two rather different fields. In this paper, by introducing two partial operations in effect algebras, we investigate the mutual relationship between involutive residuated lattices and lattice effect algebras. We prove that a lattice effect algebra under certain conditions can be extended to an involutive residuated lattice and the latter with certain properties can be restricted to the former. Especially, an sufficient and necessary condition for an involutive residuated lattice to be a lattice effect algebra with the Riesz decomposition property is obtained.

Extensions of partially ordered partial abelian monoids

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.

Holland’s theorem for pseudo-effect algebras

We give two variations of the Holland representation theorem for l-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo MV-algebra can be represented as a pseudo-effect algebra or as a pseudo MV-algebra of automorphisms of some antilattice or of some linearly ordered set.

Positive bases in ordered subspaces with the Riesz decomposition property

In this article we suppose that $E$ is an ordered Banach space whose positive cone is defined by a countable family $\mathcal F = \{f_i\mid i\in \mathbb{N}\}$ of positive continuous linear functionals on $E$, i.e. $E_+ = \{x\in E\mid f_i(x)\g

Loomis–Sikorski representation of monotone [formula omitted]-complete effect algebras

We show that monotone σ -complete effect algebras with the Riesz decomposition property are σ -homomorphic images of effect-tribes with the Riesz decomposition property, which are effect algebras of fuzzy sets closed under pointwise limits of nondecreasing fuzzy sets.

Partial algebras for Łukasiewicz logics and its extensions

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing l-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'△-, and ŁΠ-logics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings.

Unital Groups and General Comparability Property

Pseudo-effect algebras are partial algebras (E;+,0,1) with a partially defined addition + which is not necessarily commutative and therefore with two complements, left and right. If they satisfy a special kind of the Riesz decomposition property, they are intervals in unital po-groups. The general comparability property in unital po-groups with strong unit (G,u), allows to compare elements of G in some intervals with Boolean ends. Such a po-group is always an l-group admitting a state. We prove that every such (G,u) is a subdirect product of linearly ordered unital po-groups.

On Positive-off-Diagonal Operators on Ordered Normed Spaces

On a normed space X ordered by a cone K we consider a continuous linear operator A from X to X of the following kind: If a positive continuous functional f attains 0 on some positive element x, then f(Ax) is greater or equal to 0. If X is a vector lattice, then such operators can be represented as sI + B, where B is a positive operator, I is the identity and s is a real number. We generalize this assertion for weaker assumptions on X, using the Riesz decomposition property.

Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras

AbstractPseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

Quotients of partial abelian monoids and the Riesz decomposition property

Partial abelian monoids (PAMs) are structures ( $ P; \bot, \oplus, 0 $ ), where $ \oplus $ is a partially defined binary operation with domain $ \bot $ , which is commutative and associative in a restricted sense, and 0 is the neutral element. PAMs with the Riesz decomposition properties and binary relations with special properties on PAMs are studied. Relations with abelian groups, dimension equivalence and K 0 for AF C*-algebras are discussed.

Product Effect Algebras

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.

Visualization of Effect Algebras by Automorphisms

We show that every effect algebra satisfying the Riesz decomposition property can be represented as an effect algebra of automorphisms of an antilattice, and every MV-algebra can be represented as an MV-algebra of automorphisms of a linearly ordered set. Such a representation enables us to visualize effect algebras by functions. This is a variation of the Holland representation theorem for l-groups and of its generalization of Glass for directed interpolation po-groups as l-groups or po-groups automorphisms of linearly ordered set or of an antilattice, respectively.

Ideals Generated by Projections and Inductive Limit $C^*$-Algebras

We introduce two classes of inductive limit C * -algebras which generalize the AH algebras: the GAH algebras (GAH stands for generalized AH) and a subclass of it, the strong GAH algebras. We give necessary and sufficientconditions for an ideal of a GAH algebra to be generated by projections which, in particular, gives necessary and sufficient conditions for a GAH algebra to have the ideal property, i.e., any ideal is generated by projections. We prove that if 0→I→A→B→0 is an exact sequence of C * -algebras such that A is a GAH algebra, then A has the ideal property if and only if I and B have the ideal property. We describe the lattice of ideals generated by projections of a strong GAH algebra and also the partially ordered set of the stably cofinite ideals generated by projections of a strong GAH algebra A under the additional assumption that the projections in M ∞ (A) satisfy the Riesz decomposition property. These results generalize some of our previous theorems involving AH algebras.

Blocks of homogeneous effect algebras

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, calledhomogeneous effect algebras. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying the Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying the Riesz decomposition property. This generalizes a recent result by Riec˘anová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalisation of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebraEforms an orthoalgebraEs. Every block ofEsis the centre of a block ofE. The set of all sharp elements in the compatibility centre ofEcoincides with the centre ofE. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space ℍ with dim (ℍ) > 1, the standard effect algebra ℰ(ℍ) of all effects in ℰ is not homogeneous.

The Riesz decomposition property for the space of regular operators

If E E and F F are Banach lattices such that E E is separable and F F has the countable interpolation property, then the space of all continuous regular operators L \mathcal {L} r ( E , F ) ^r(E,F) has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.