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Basic Operational Preorders for Algebraic Effects in General, and for Combined Probability and Nondeterminism in Particular

2018
*
Annual Conference for Computer Science Logic
*

The "

doi:10.4230/lipics.csl.2018.29
dblp:conf/csl/LopezS18
fatcat:wins7aui3bgvbkkwwv3hpp4z3e
*generic**operational*metatheory" of Johann, Simpson*and*Voigtländer (LiCS 2010) defines contextual equivalence, in the presence of*algebraic**effects*, in terms of a basic*operational*preorder on ground-type ... We propose three*general*approaches to specifying such preorders: (i)*operational*(ii) denotational,*and*(iii) axiomatic; coinciding with the three major styles of program semantics. ... Acknowledgements We thank Gordon Plotkin, Matija Pretnar*and*Niels Voorneveld for helpful discussions. ...##
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Intervals in Generalized Effect Algebras and their Sub-generalized Effect Algebras

2013
*
Acta Polytechnica
*

We give a condition on E

doaj:128bdb203f0241feb29461f1f4251ef9
fatcat:o5dfho26xna2hb4c7w7pitkn74
*and*G under which every such G is a sub-*generalized**effect**algebra*of E. ... We consider subsets G of a*generalized**effect**algebra*E with 0∈G*and*such that every interval [0, q]G = [0, q]E ∩ G of G (q ∈ G , q ≠ 0) is a sub-*effect**algebra*of the*effect**algebra*[0, q]E. ... Acknowledgements This work was supported by grants VEGA 1/0297/11*and*VEGA 1/0426/12 of the Ministry of Education of the Slovak Republic*and*by grant APVV-0178-11 of the Slovak Research*and*Development ...##
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Extensions of Effect Algebra Operations

2011
*
Acta Polytechnica
*

Further, sub-

doaj:78e9675b10884358a2903f9c74794f24
fatcat:dqsk7x4xhnh2hkjj5gj66kgm5y
*generalized**effect**algebras**and*interval*effect**algebras*with respect of these*operations*are investigated. ... We equip this set with various*effect**algebraic**operations*making it a*generalized**effect**algebra*. ... Acknowledgement Supported by grants VEGA 1/0297/11*and*VEGA 1/0021/10 of the Ministry of Education of the Slovak Republic. ...##
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MAXIMAL SUBSETS OF PAIRWISE SUMMABLE ELEMENTS IN GENERALIZED EFFECT ALGEBRAS

2013
*
Acta Polytechnica
*

Keywords: (

doi:10.14311/ap.2013.53.0457
fatcat:hxsea5t2orc53nggap3ngeutyy
*generalized*)*effect**algebra*, MV-*effect**algebra*, summability block, compatibility block, linear*operators*in Hilbert spaces. ... 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. ... Acknowledgements Zdenka Riečanová gratefully acknowledges support by the Science*and*Technology Assistance Agency under contract APVV-0178- 11 ...##
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Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

2011
*
Reports on mathematical physics
*

We show that (

doi:10.1016/s0034-4877(12)60009-3
fatcat:d7wylz2rwrbida354dspmlarsq
*generalized*)*effect**algebras*may be suitable very simple*and*natural*algebraic*structures for sets of (unbounded) positive self-adjoint linear*operators*densely defined on an infinite-dimensional ... In these cases the*effect**algebraic**operation*, as a total or partially defined binary*operation*, coincides with the usual addition of*operators*in Hilbert spaces. ... Then (F (H); ⊕, 0) is a*generalized**effect**algebra*. have shown that V(H)*and*S p (H) are*generalized**effect**algebras*under the partial*operations*⊕*and*⊕ S , respectively. ...##
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On Realization of Generalized Effect Algebras

2012
*
Reports on mathematical physics
*

This extends the corresponding results for

doi:10.1016/s0034-4877(12)60052-4
fatcat:g7zm6q6seng2xj4dcfiyz3pxni
*effect**algebras*of Riečanová*and*Zajac. Further, any*operator**generalized**effect**algebra*G_D( H) possesses an order determining set of*generalized*states. ... We show that a*generalized**effect**algebra*is representable in the*operator**generalized**effect**algebra*G_D( H) of*effects*of a complex Hilbert space H iff it has an order determining set of*generalized*... In*general*form, an*effect**algebra*is in fact a partial*algebra*with one partial binary*operation**and*two unary*operations*satisfying the following axioms due to Foulis*and*Bennett. ...##
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On Bilinear Forms from the Point of View of Generalized Effect Algebras

2013
*
Foundations of physics
*

We show when different families of bilinear forms can be described as a

doi:10.1007/s10701-013-9736-2
fatcat:mj26pjbpkfhnjcpovwnip5bxni
*generalized**effect**algebra*. ... In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete*generalized**effect**algebras*. ... Elements of*generalized**effect**algebras*In the section we gather some basic definitions*and*facts on*generalized**effect**algebras**and**effect**algebras*. ...##
###
Musings about the Triple Representation Theorem for Effect Algebras

2012
*
Order
*

*Generalized*

*effect*

*algebras*Definition (1) A

*generalized*

*effect*

*algebra*(E; ⊕, 0) is a set E with element 0 ∈ E

*and*partial binary

*operation*⊕ satisfying, for any x, y, z ∈ E, conditions (GE1) x ⊕ y ...

*Generalized*

*effect*

*algebras*Definition (1) A

*generalized*

*effect*

*algebra*(E; ⊕, 0) is a set E with element 0 ∈ E

*and*partial binary

*operation*⊕ satisfying, for any x, y, z ∈ E, conditions (GE1) x ⊕ y ... Let T(E) be a subset of S(E) × M(E) given by Equip T(E) with a partial binary

*operation*Theorem Let E be an

*effect*

*algebra*satisfying (C). ...

##
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A note on unitizations of generalized effect algebras

2015
*
Soft Computing - A Fusion of Foundations, Methodologies and Applications
*

There is a forgetful functor from the category of

doi:10.1007/s00500-015-1755-9
fatcat:57xlrivra5gtjm35dyeomjkmv4
*generalized**effect**algebras*to the category of*effect**algebras*. ... We prove that this functor is a right adjoint*and*that the corresponding left adjoint is the well-known unitization construction by Hedl\'ikov\'a*and*Pulmannov\'a. ... Acknowledgements This research is supported by grant VEGA G-2/0059/12 of MŠ SR, Slovakia*and*by the Slovak Research*and*Development Agency under the contracts APVV-0073-10, APVV-0178-11. ...##
###
Page 5726 of Mathematical Reviews Vol. , Issue 2004g
[page]

2004
*
Mathematical Reviews
*

They serve as noncommutative

*generalizations*of*effect**algebras*,*and*hence as noncommutative*generalizations*of both orthomodular posets*and*MV-*algebras*. ...*Effect**algebras*, as partial*algebraic*structures with a partially de- fined*operation*+*and*constants 0*and*1, have been introduced as an abstraction of Hilbert space*effects*, i.e. self-adjoint opera- ...##
###
On Realization of Partially Ordered Abelian Groups

2012
*
International Journal of Theoretical Physics
*

Since every (

doi:10.1007/s10773-012-1426-x
fatcat:ivpq6rv65vfltbspwr74a5oopu
*generalized*)*effect**algebra*with an order determining set of (*generalized*) states can be represented by means of an abelian partially ordered group*and*events in quantum mechanics can be described ... We also characterize*effect**algebras*with an order determining set of states as interval*operator**effect**algebras*in groups of self-adjoint bounded linear*operators*. ... Since every (*generalized*)*effect**algebra*with an order determining set of (*generalized*) states can be represented by means of an abelian partially ordered group*and*events in quantum mechanics can be described ...##
###
MV-pairs and state operators

2015
*
Fuzzy sets and systems (Print)
*

Flaminio, Montagna, 2009 state

doi:10.1016/j.fss.2014.04.002
fatcat:r7sq6qfyknd6lpk7z7jqygs7si
*operator*on MV-*algebras*-unary*operation*with basic properties of statesstate MV-*algebras*.*Effect**algebras*Foulis, Bennett, 1994-modelling unsharp measurements in QM. ... We study relations between state*operators**and*MV-pairs. MV-pairs*and*state*operators*... =⇒ ∃ a unique Boolean*algebra*B(M ) -R-*generated*Boolean*algebra*-such that M is a 0, 1-sublattice*and*M*generates*B(M ) as a boolean ring. ...##
###
Page 7632 of Mathematical Reviews Vol. , Issue 2004j
[page]

2004
*
Mathematical Reviews
*

(English summary)

*General**algebra**and*ordered sets. Tatra Mt. Math. Publ. 27 (2003), 45-65. A theory of ideals of pseudo-*effect**algebras*is developed. ... Pseudo-*effect**algebras*were introduced by the author*and*Vetterlein as noncommutative*generalizations*of*effect**algebras*which play an important role in the mathematical formulation of quantum me- chanical ...##
###
Closure systems over effect algebras

2021
*
Nexo
*

At first, we will define closure systems over

doi:10.5377/nexo.v34i02.11558
fatcat:eaoav6m5i5csvkshuei5bn53r4
*effect**algebras*,*and*for arbitrary set $ U $*and*arbitrary subset S of all functions from U to an*effect**algebra*L we will obtain the closure system containing ... Then, we will define the base of this closure system,*and*for arbitrary subset S of all functions from U to an*effect**algebra*L we will obtain the base of this closure system. ... The direct product of family { : ∈ } of*effect**algebras*, denoted by ∏ ∈ is an*effect**algebra*in which partially binary*operation*+*and*unary*operation*(−) ´ defined pointwise. ...##
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A Diagrammatic Calculus for Algebraic Effects
[article]

2020
*
arXiv
*
pre-print

We give a formal foundation for such a calculus in terms of Lawvere theories

arXiv:2001.01337v2
fatcat:xql67jnewrdk5pihzbcrx5hgua
*and**generic**effects*. ... Additionally, we show a number of*algebraic**and*order-theoretic laws on diagrams, this way laying the foundations for a diagrammatic calculus of*algebraic**effects*. ... We have introduced a diagrammatic notation*and*calculus for countable monads with*algebraic**operations*. We have shown by means of examples*and**general*results some of advantages of our notation. ...
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