# Jiří JandaMasaryk University | MUNI

Jiří Janda

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17

Publications

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35

Citations

## Publications

Publications (17)

We showed that the set of observables O(E) on a σ-frame effect algebra E is in one-to-one correspondence with the set USC(is) of point-free versions of real upper semicontinous functions on the order reduct of E. The Olson order and the sum on O(E) are represented by the usual order and the sum of real upper semicontinuous functions. Sharp observab...

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 \...

Group coextensions of monoids, which generalise Schreier-type extensions of groups, have originally been defined by P.A. Grillet and J. Leech. The present paper deals with pomonoids, that is, monoids that are endowed with a compatible partial order. Following the lines of the unordered case, we define pogroup coextensions of pomonoids. We furthermo...

The natural question about the sum of observables on \(\sigma \)-complete MV-effect algebras, which was recently defined by A. Dvurečenskij, is how it affects spectra of observables, particularly, their extremal points. We describe boundaries for extremal points of the spectrum of the sum of observables in a general case, and we give necessary and...

The existence of a non-trivial singular positive bilinear form Simon (J. Funct. Analysis 28, 377–385 (1978)) yields that on an infinite-dimensional complex Hilbert space \({\mathcal {H}}\) the set of bilinear forms \({\mathcal {F}}(\mathcal {H})\) is richer than the set of linear operators \({\mathcal {V}}(\mathcal {H})\). We show that there exists...

We study measures, finitely additive measures, regular measures, and
$\sigma$-additive measures that can attain even infinite values on the quantum
logic of a Hilbert space. We show when particular classes of non-negative
measures can be studied in the frame of generalized effect algebras.

Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra \( E\) and how to compute all possible pairs of tense operators on \( E\) . However, we firstly need to derive a time frame which enables...

We study the so-called pairwise summable generalized effect algebras and show some conditions under which they are generalized MV-effect algebras. Moreover, we give a necessary and sufficient condition for finite antichains of elements in pairwise summable generalized effect algebras under which they are sets of atoms of sub-generalized effect alge...

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritanc...

We show that in any generalized effect algebra (G;⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G; ⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by...

We study positive bilinear forms on a Hilbert space which are not
necessarily bounded nor induced by some positive operator. We show when
different families of bilinear forms can be described as a generalized
effect algebra. In addition, we present families which are or are not
monotone downwards (Dedekind upwards) σ-complete generalized
effect alg...

In (Riečanová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set
$\mathcal{M}$
of states can by embedded into a Hilbert space effect algebra
$\mathcal{E}(l_{2}(\mathcal{M}))$
. We consider the problem when its effect algebraic MacNeille completion
$\hat{E}$
can be also embedded into t...

The notion of a generalized effect algebra is presented as a generalization of effect algebra for an algebraic description of the structure of the set of all positive linear operators densely defined on a Hilbert space with the usual sum of operators. The structure of the set of not only positive linear operators can be described with the notion of...

Tense operators for MV-algebras were introduced by Diaconescu and Georgescu. Based on their definition Chajda and Kolařík presented the definition of tense operators for lattice effect algebras. Chajda and Paseka tackled the problem of axiomatizing tense operators on an effect algebra by introducing the notion of a partial dynamic effect algebra. T...

The generalized effect algebra was presented as a generalization of effect algebra for an algebraic description of the structure of the set of all positive linear operators densely defined on Hilbert space with the usual sum of operators. A structure of the set of not only positive linear operators can be described with the notion of weakly ordered...

We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space H. In [Paseka, J.– –Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65–72] there is introduced the notion of a weakly ordered partial commutative group and sh...

We continue in the direction of our paper on PT -Symmetry in (Generalized) Effect Algebras and Partial Groups. Namely we extend our considerations to the setting of weakly ordered partial groups. In this setting, any operator weakly ordered partial group is a pasting of its partially ordered commutative subgroups of linear operators with a fixed de...