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

2017
*
arXiv
*
pre-print

Moisil

arXiv:1710.02393v1
fatcat:yey4rlgdgvcg5hwywyhnehbe5u
*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*. ...##
###
Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets
[article]

2019
*
arXiv
*
pre-print

By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene

arXiv:1806.09203v3
fatcat:ltetwjitgfhsdoay4k74l6b4by
*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. ...##
###
Reasoning About Algebraic Structures with Implicit Carriers in Isabelle/HOL
[chapter]

2020
*
Lecture Notes in Computer Science
*

We prove Chen and Grätzer's construction theorem for

doi:10.1007/978-3-030-51054-1_14
fatcat:7oikixg6rrdhdeiwl3j7k64fty
*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*. ...##
###
Ultrafilter Extensions for Coalgebras
[chapter]

2005
*
Lecture Notes in Computer Science
*

This paper studies finitary modal logics as specification languages for

doi:10.1007/11548133_17
fatcat:bxsfjcbgwfbcnchuzmq3sxzqmi
*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 :*Stone*→*Stone*. ...##
###
Stone Coalgebras

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. ...

##
###
Aleph–zero categorical Stone algebras

1978
*
Journal of the Australian Mathematical Society
*

This paper is a contribution to the problem of characterizing the K 0 -categorical

doi:10.1017/s1446788700011861
fatcat:fugjv52zpvhsja772gurpxnvbq
*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*. ...##
###
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

1997
*
Theoretical Computer Science
*

The collection of all subsets of a

doi:10.1016/s0304-3975(96)00334-9
fatcat:huvo5vrdjzatzfwzzivr7qsmze
*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 ...##
###
Stone coalgebras

2004
*
Theoretical Computer Science
*

We argue that the category of

doi:10.1016/j.tcs.2004.07.023
fatcat:st3xvxtfo5hw3pqlh3tst2bise
*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. ...##
###
On connections between information systems, rough sets and algebraic logic

1993
*
Banach Center Publications
*

Pomyka la [11] that the collection of rough

doi:10.4064/-28-1-117-124
fatcat:dl4trov2bnemvaigz5fpebaelq
*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*. ...##
###
Unrestricted stone duality for Markov processes

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. ...

##
###
Stone Duality for Markov Processes

2013
*
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
*

We

doi:10.1109/lics.2013.38
dblp:conf/lics/KozenLMP13
fatcat:dix26nhntvfy7p56uv65gpa5ci
*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 . ...##
###
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. ...##
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A Coalgebraic Approach to Dualities for Neighborhood Frames
[article]

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 . ...

##
###
Stone pseudovarieties
[article]

2021
*
arXiv
*
pre-print

This provides a new approach to duality theory which,

arXiv:1910.03674v2
fatcat:ujihyeed3regxbftkprxhpq2g4
*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. ...
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