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

Aliaume Lopez, Alex Simpson, Michael Wagner
2018 Annual Conference for Computer Science Logic  
The "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.  ... 
doi:10.4230/lipics.csl.2018.29 dblp:conf/csl/LopezS18 fatcat:wins7aui3bgvbkkwwv3hpp4z3e

Intervals in Generalized Effect Algebras and their Sub-generalized Effect Algebras

Zdenka Riečanová, Michal Zajac
2013 Acta Polytechnica  
We give a condition on E 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  ... 
doaj:128bdb203f0241feb29461f1f4251ef9 fatcat:o5dfho26xna2hb4c7w7pitkn74

Extensions of Effect Algebra Operations

Z. Riečanová, M. Zajac
2011 Acta Polytechnica  
Further, sub-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.  ... 
doaj:78e9675b10884358a2903f9c74794f24 fatcat:dqsk7x4xhnh2hkjj5gj66kgm5y

MAXIMAL SUBSETS OF PAIRWISE SUMMABLE ELEMENTS IN GENERALIZED EFFECT ALGEBRAS

Zdenka Riečanová, Jiří Janda
2013 Acta Polytechnica  
Keywords: (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  ... 
doi:10.14311/ap.2013.53.0457 fatcat:hxsea5t2orc53nggap3ngeutyy

Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

Z. Riečanová, M. Zajac, S. Pulmannová
2011 Reports on mathematical physics  
We show that (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 operationsand ⊕ S , respectively.  ... 
doi:10.1016/s0034-4877(12)60009-3 fatcat:d7wylz2rwrbida354dspmlarsq

On Realization of Generalized Effect Algebras

Jan Paseka
2012 Reports on mathematical physics  
This extends the corresponding results for 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.  ... 
doi:10.1016/s0034-4877(12)60052-4 fatcat:g7zm6q6seng2xj4dcfiyz3pxni

On Bilinear Forms from the Point of View of Generalized Effect Algebras

Anatolij Dvurečenskij, Jiří Janda
2013 Foundations of physics  
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 algebras.  ...  Elements of generalized effect algebras In the section we gather some basic definitions and facts on generalized effect algebras and effect algebras.  ... 
doi:10.1007/s10701-013-9736-2 fatcat:mj26pjbpkfhnjcpovwnip5bxni

Musings about the Triple Representation Theorem for Effect Algebras

Josef Niederle, Jan Paseka
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).  ... 
doi:10.1007/s11083-012-9259-z fatcat:nddadvdurrgmjgtymqvzzjsq6q

A note on unitizations of generalized effect algebras

Gejza Jenča
2015 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
There is a forgetful functor from the category of 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.  ... 
doi:10.1007/s00500-015-1755-9 fatcat:57xlrivra5gtjm35dyeomjkmv4

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

Ivan Chajda, Jan Paseka, Lei Qiang
2012 International Journal of Theoretical Physics  
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  ...  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  ... 
doi:10.1007/s10773-012-1426-x fatcat:ivpq6rv65vfltbspwr74a5oopu

MV-pairs and state operators

S. Pulmannová, E. Vinceková
2015 Fuzzy sets and systems (Print)  
Flaminio, Montagna, 2009 state 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.  ... 
doi:10.1016/j.fss.2014.04.002 fatcat:r7sq6qfyknd6lpk7z7jqygs7si

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

Mahdi Ronasi, Esfandiar Eslami
2021 Nexo  
At first, we will define closure systems over 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.  ... 
doi:10.5377/nexo.v34i02.11558 fatcat:eaoav6m5i5csvkshuei5bn53r4

A Diagrammatic Calculus for Algebraic Effects [article]

Ugo Dal Lago, Francesco Gavazzo
2020 arXiv   pre-print
We give a formal foundation for such a calculus in terms of Lawvere theories 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.  ... 
arXiv:2001.01337v2 fatcat:xql67jnewrdk5pihzbcrx5hgua
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