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Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics [article]

Arun Kumar
2017 arXiv   pre-print
Moisil in 1941, while constructing the algebraic models of n-valued Łukasiewicz logic defined the set B^[n],where B is a Boolean algebra and 'n' being a natural number.  ...  ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and 4-valued semantics of logic for double Stone algebras are proposed and established soundness and completeness  ...  In Section 2, we provide structural representations of Stone and dual Stone algebras in which elements of these algebra are defined by pairs of Boolean elements and rough sets.  ... 
arXiv:1710.02393v1 fatcat:yey4rlgdgvcg5hwywyhnehbe5u

Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets [article]

Jouni Järvinen, Sándor Radeleczki
2019 arXiv   pre-print
By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene algebra is isomorphic to a subalgebra of the rough set regular pseudocomplemented Kleene algebra defined by a tolerance  ...  We also characterize the Kleene-Varlet spaces corresponding to the regular pseudocomplemented Kleene algebras satisfying the Stone identity.  ...  set regular double Stone algebra defined by E.  ... 
arXiv:1806.09203v3 fatcat:ltetwjitgfhsdoay4k74l6b4by

Reasoning About Algebraic Structures with Implicit Carriers in Isabelle/HOL [chapter]

Walter Guttmann
2020 Lecture Notes in Computer Science  
We prove Chen and Grätzer's construction theorem for Stone algebras in Isabelle/HOL.  ...  The development requires extensive reasoning about algebraic structures in addition to reasoning in algebraic structures.  ...  The theory HOL/Algebra/Ideal.thy defines ring-theoretic ideals in locales with a carrier set. In the theory HOL/Filter.thy a filter is defined as a set of sets.  ... 
doi:10.1007/978-3-030-51054-1_14 fatcat:7oikixg6rrdhdeiwl3j7k64fty

Ultrafilter Extensions for Coalgebras [chapter]

C. Kupke, A. Kurz, D. Pattinson
2005 Lecture Notes in Computer Science  
This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality.  ...  This leads us to study the relationship of finitary modal logics and Set-coalgebras by uncovering the relationship between Set-coalgebras and Stone-coalgebras.  ...  The previous definition pre-supposes a set Λ of predicate liftings to define the lifted functorT : StoneStone.  ... 
doi:10.1007/11548133_17 fatcat:bxsfjcbgwfbcnchuzmq3sxzqmi

Stone Coalgebras

Clemens Kupke, Alexander Kurz, Yde Venema
2003 Electronical Notes in Theoretical Computer Science  
In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor.  ...  We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way.  ...  Acknowledgments We would like to thank the participants of the ACG-meetings at CWI, in particular, Marcello Bonsangue, Alessandra Palmigiano, and Jan Rutten.  ... 
doi:10.1016/s1571-0661(04)80638-8 fatcat:m2fs3hoggrcl7giex2amwykxim

Aleph–zero categorical Stone algebras

Philip Olin
1978 Journal of the Australian Mathematical Society  
This paper is a contribution to the problem of characterizing the K 0 -categorical Stone algebras.  ...  If the dense set is a Boolean algebra, we show that this type of reduction works for certain subclasses but not for all such algebras.  ...  The remaining results in this paper are concerned with Stone algebras whose dense set is a Boolean algebra.  ... 
doi:10.1017/s1446788700011861 fatcat:fugjv52zpvhsja772gurpxnvbq

Page 1339 of Mathematical Reviews Vol. 43, Issue 6 [page]

1972 Mathematical Reviews  
If S is a Boolean algebra on a set B (a Boolean algebra regarded as special Stone algebra) then the sublattice of SxS defined on {<a, b>|a,b¢ B,a<b} as well as its dual is pseudo-complemented, and this  ...  If, in addition, there exists a unary operation * on S such that <S; v, a, *, 1, 0> is a Stone algebra, © is called a double Stone algebra. Let C(G) be the set of all complemented elements of S.  ... 

A logic for rough sets

Ivo Düntsch
1997 Theoretical Computer Science  
The collection of all subsets of a set forms a Boolean algebra under the usual set-theoretic operations, while the collection of rough sets of an approximation space is a regular double Stone algebra (  ...  Using the representation theorem for these algebras by Katri%k (1974), we present such a logic for rough sets and its algebraic semantics in the spirit of Andrtka and NCmeti (1994).  ...  Acknowledgements I thank the referees for their constructive remarks and the pointers to earlier literature, Hajnal Andrkka for providing the information regarding the history of the algebraic semantics  ... 
doi:10.1016/s0304-3975(96)00334-9 fatcat:huvo5vrdjzatzfwzzivr7qsmze

Stone coalgebras

Clemens Kupke, Alexander Kurz, Yde Venema
2004 Theoretical Computer Science  
We argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category  ...  Dually, we define a T-algebra to be a T op -coalgebra and Alg(T ) = (Coalg(T op )) op . Example 2.2. A Kripke frame is a structure F = (X, R) such that R is a binary relation on X.  ...  Special thanks are due to one of the anonymous referees who provided us with very interesting comments, some of which have been incorporated in Remarks 4.5, 4.15 and 5.9.  ... 
doi:10.1016/j.tcs.2004.07.023 fatcat:st3xvxtfo5hw3pqlh3tst2bise

On connections between information systems, rough sets and algebraic logic

Stephen Comer
1993 Banach Center Publications  
Pomyka la [11] that the collection of rough sets of an approximation space forms a Stone algebra.  ...  algebras, relation algebras, and Stone algebras.  ...  In a similar way relativized algebras of other types of systems can be defined.  ... 
doi:10.4064/-28-1-117-124 fatcat:dl4trov2bnemvaigz5fpebaelq

Unrestricted stone duality for Markov processes

Robert Furber, Dexter Kozen, Kim Larsen, Radu Mardare, Prakash Panangaden
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
Stone duality relates logic, in the form of Boolean algebra, to spaces.  ...  In particular, we do not require that the probabilistic modalities preserve a distinguished base of clopen sets, nor that morphisms of Markov processes do so.  ...  The analogous functor in Stone duality takes the Boolean algebra of clopens of a Stone space.  ... 
doi:10.1109/lics.2017.8005152 dblp:conf/lics/FurberKLMP17 fatcat:q2azv5bdubf3dbzqhurcms27ee

Stone Duality for Markov Processes

Dexter Kozen, Kim G. Larsen, Radu Mardare, Prakash Panangaden
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions.  ...  We prove a Stone-type duality theorem between countable Aumann algebras and countably-generated continuous-space Markov processes.  ...  The categories of Aumann algebras (AA) and Stone Markov processes (SMP) were defined in §4 and §5, respectively. We define contravariant functors A : SMP → AA op and M : AA → SMP op .  ... 
doi:10.1109/lics.2013.38 dblp:conf/lics/KozenLMP13 fatcat:dix26nhntvfy7p56uv65gpa5ci

Page 1461 of Mathematical Reviews Vol. 47, Issue 6 [page]

1974 Mathematical Reviews  
If one denotes the set of dense elements of a Stone lattice L by D(L) then D(P), D?(P), ---, D®-1(P)=1 are also relative Stone algebras. A Stone algebra of order n, n2 2, is now defined as follows.  ...  ordered set of prime ideals of a Stone algebra of order n.  ... 

A Coalgebraic Approach to Dualities for Neighborhood Frames [article]

Guram Bezhanishvili, Nick Bezhanishvili, Jim de Groot
2022 arXiv   pre-print
In the first part of the paper we construct an endofunctor on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on 𝖲𝖾𝗍.  ...  Using one-step axioms in the language of finitary modal logic, we restrict this duality to other classes of neighborhood algebras studied in the literature, including monotone modal algebras and contingency  ...  In the other direction, clp is the functor that sends a Stone space to its Boolean algebra of clopen sets and a continuous function f to f −1 .  ... 
arXiv:2106.01628v2 fatcat:w4rxxjcj4ne5bn3wjanuoubn7e

Stone pseudovarieties [article]

Jorge Almeida, Ondřej Klíma
2021 arXiv   pre-print
This provides a new approach to duality theory which, in the case of a Stone signature, culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone  ...  Profinite algebras are the residually finite compact algebras, whose underlying topological spaces are Stone spaces.  ...  Nevertheless, of course every set Σ of identities still defines a Stone variety, namely the class of all Stone topological algebras that satisfy Σ.  ... 
arXiv:1910.03674v2 fatcat:ujihyeed3regxbftkprxhpq2g4
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