Elena Vinceková

Sharp and Fuzzy Observables on Effect Algebras

Abstract Observables on effect algebras and their fuzzy versions obtained by means of confidence measures (Markov kernels) are studied. It is shown that, on effect algebras with the (E)-property, given an observable and a confidence measure, there exists a fuzzy version of the observable. Ordering of observables according to their fuzzy properties is introduced, and some minimality conditions with respect to this ordering are found. Applications of some results of classical theory of experiments are considered.

Remarks on the order for quantum observables

Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.

MV-pairs and state operators

Flaminio and Montagna (2008) enlarged the language of MV-algebras by a unary operation ?, called internal state or state operator, equationally defined so as to preserve the basic properties of a state in its usual meaning. The resulting class of MV-algebras is called state MV-algebras. Jenca (2007) and Vetterlein (2008), using different approaches, represented MV-algebras through the quotient of a Boolean algebra B by a suitable subgroup G of the group of all automorphisms of B. Such a couple ( B , G ) is called an MV-pair. We introduce the notion of a state MV-pair as a triple ( B , G , ? ) , where ( B , G ) is an MV-pair and ? is a state operator on B, and show that there are relations between state MV-pairs and state MV-algebras similar to the relations between MV-pairs and MV-algebras. We also give a characterization of those MV-pairs, resp. state MV-pairs, that induce subdirectly irreducible MV-algebras, resp. state MV-algebras.

Abelian extensions of partially ordered partial monoids

In this paper we introduce the notion of a partially ordered partial monoid (po PM), and we study extensions of cancellative po PMs by abelian groups. We concentrate on the so-called central extensions, and prove that every such extension is an F-product of a po PM by an abelian group defined by a cocycle

Lattice pseudoeffect algebras as double residuated structures

f.

Unitizations of Generalized Pseudo Effect Algebras and their Ideals

Like in partially ordered groups, the extension can be ordered by means of special sets. We also compare extensions of po PMs with the extensions of their universal groups.

State-morphism pseudo-effect algebras

An MV-pair is a BG-pair (B, G) (where B is a Boolean algebra and G is a subgroup of the automorphism group of B) satisfying certain conditions. Recently, it was proved by Jenča that, given an MV-pair (B, G), the quotient B/~G, where ~G is an equivalence relation naturally associated with G, is an MV-algebra, and conversely, to every MV-algebra there corresponds an MV-pair. In this paper, we study relations between congruences of B and congruences of B/~G induced by a G-invariant ideal I of B. In addition we bring some relations between ideals in MV-algebras and in the corresponding R-generated Boolean algebras.

Congruences and ideals in pseudo effect algebras as total algebras

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.