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Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

G. Cattaneo, D. Ciucci, D. Dubois
2011 Information Sciences  
The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N , K,  ...  Finally, the question of the role of these deviant modalities, as opposed to the usual nondistributive ones, in the scope of knowledge representation and approximation spaces is discussed.  ...  always based on a residuated lattice.  ... 
doi:10.1016/j.ins.2011.05.008 fatcat:k4qgbd3o2jgs5or2flwanzqjw4

Incomplete Information: Structure, Inference, Complexity

Jouni Järvinen
2006 Studia Logica: An International Journal for Symbolic Logic  
Pawlak has also introduced the notion of rough set approximations which naturally arise from information systems.  ...  Because the correspondence between equivalences and partitions is one-to-one, every indiscernibility relation induces a partition of the c  ...  In the book, fuzzy information systems are such that the values of attributes are L-fuzzy subsets, where L is a double residuated lattice, that is, a lattice with additional operators of product, sum,  ... 
doi:10.1007/s11225-006-9018-5 fatcat:et7azj3uhrfhjm6vacixqamfne

Hidden modalities in algebras with negation and implication

JOUNI JARVINEN, MICHIRO KONDO, JORMA K. MATTILA, SANDOR RADELECZKI
2013 Mathematics for applications  
We define analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds.  ...  Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5).  ...  They are distributive lattices with a De Morgan negation and n−1 modal operators fulfilling the axioms given in Definition 3.2 defined on them.  ... 
doi:10.13164/ma.2013.02 fatcat:kkruc3saa5aajlwqhij2disski

Dynamical Systems On Weighted Lattices: General Theory [article]

Petros Maragos
2017 arXiv   pre-print
The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra.  ...  We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using  ...  This research was supported by the project "COGNIMUSE" under the "ARISTEIA" Action of the Operational Program Education and Lifelong Learning and was co-funded by the European Social Fund and Greek National  ... 
arXiv:1606.07347v4 fatcat:fq7px7ex5vd3pddvg2zulldgxm

Extending conceptualisation modes for generalised Formal Concept Analysis

Francisco J. Valverde-Albacete, Carmen Peláez-Moreno
2011 Information Sciences  
Formal Concept Analysis (FCA) is an exploratory data analysis technique for boolean relations based on lattice theory.  ...  In this paper we augment this formalism in two ways: first we extend FCA to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal-logic motivated way.  ...  We would like to thank M. Erné for providing initial input for Galois connection diagrams in section 2.1.1. We would also like to thank the reviewers of early versions of this paper.  ... 
doi:10.1016/j.ins.2010.04.014 fatcat:kdstbuyex5attiyvtq42bmkify

Modal twist-structures over residuated lattices

H. Ono, U. Rivieccio
2013 Logic Journal of the IGPL  
We define a logic that corresponds to our twist-structures and show how to expand it with modal operators, obtaining a paraconsistent many-valued modal logic that generalizes existing work on modal expansions  ...  We introduce a class of algebras, called twist-structures, whose members are built as special squares of an arbitrary residuated lattice.  ...  residuated lattice (with modal operators).  ... 
doi:10.1093/jigpal/jzt043 fatcat:5duui4gcvzhbpbp6az5lfmgrti

Dynamical systems on weighted lattices: general theory

Petros Maragos
2017 MCSS. Mathematics of Control, Signals and Systems  
The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra.  ...  We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using  ...  On complete lattices an increasing operator ψ is residuated (resp. a residual ψ ) if and only if it is a dilation (resp. erosion).  ... 
doi:10.1007/s00498-017-0207-8 fatcat:cff5jfdvpfbz7khca5lhhqumbi

Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments [chapter]

Didier Dubois, Francesc Esteva, Lluís Godo, Henri Prade
2007 Handbook of the History of Logic  
and modalities emerged.  ...  The idea of fuzzy sets introduced in the early sixties [Zadeh, 1965] and the development of fuzzy logic later on [Zadeh, 1975a] has brought forward a new formal framework for capturing graded imprecision  ...  residuated lattices by Medina et al. [2001] .  ... 
doi:10.1016/s1874-5857(07)80009-4 fatcat:s4i5zfomgzbxnfbuif5ffsesea

L-fuzzy quantifiers of type determined by fuzzy measures

Antonín Dvořák, Michal Holčapek
2009 Fuzzy sets and systems (Print)  
The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy set, where L is a complete residuated lattice.  ...  In this contribution, fuzzy measures are defined on algebras of fuzzy sets (measure spaces) and, generally, they attain values from a complete residuated lattice L.  ...  A divisible residuated lattice satisfying the law of double negation is called an MV-algebra. For other information about residuated lattices we refer to [2, 23] . Example 2.1.  ... 
doi:10.1016/j.fss.2009.05.010 fatcat:yp6wamvlbfhdpchhbvwqkyryqi

Fuzzy Sets: History and Basic Notions [chapter]

Didier Dubois, Walenty Ostasiewicz, Henri Prade
2000 Fundamentals of Fuzzy Sets  
This Chapter also contains a discussion on operational semantics of the generally too abstract notion of membership function.  ...  This paper is an introduction to fuzzy set theory. It has several purposes. First, it tries to explain the emergence of fuzzy sets from an historical perspective.  ...  In fact, besides the modality possible, Aristotle already considered other so-called modal expressions like impossible, and necessary, giving the base for the so-called modal logic, for which the negation  ... 
doi:10.1007/978-1-4615-4429-6_2 fatcat:sboduqw2l5a4beutph36bfynku

Kripke Semantics for Modal Bilattice Logic

Achim Jung, Umberto Rivieccio
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based  ...  on Belnap's four-valued logic.  ...  ACKNOWLEDGEMENTS We would like to thank the authors of [6] for providing the inspiration for the current study.  ... 
doi:10.1109/lics.2013.50 dblp:conf/lics/JungR13 fatcat:zamz46j52nhj7dvfovd5aoqk4a

Three-Valued Logics, Uncertainty Management and Rough Sets [chapter]

Davide Ciucci, Didier Dubois
2014 Lecture Notes in Computer Science  
Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such  ...  We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an  ...  From Three-Valued Rough Set Logic to Modal Logic We have seen two different modal approaches concerning rough sets: a Boolean one based on S5 and a many-valued one, the PRL logic, the latter being based  ... 
doi:10.1007/978-3-642-54756-0_1 fatcat:o7vun6jnzncxrizq3vfwxvji34

Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations

M. Gehrke, H.A. Priestley
2007 Journal of Pure and Applied Algebra  
The context for this paper is a class of distributive lattice expansions, called double quasioperator algebras (DQAs).  ...  In many cases of interest, binary residuated operations are present. An operation h which, coordinatewise, preserves ∨ and 0 lifts to an operation which is residuated, even when h is not.  ...  Acknowledgments The authors would like to thank the referee for a careful reading of the original version of this paper and for constructive comments.  ... 
doi:10.1016/j.jpaa.2006.06.001 fatcat:2h4imvl5h5cwreydtbowgyio6q

Monadic NM-algebras [article]

Jun Tao Wang, Xiao Long Xin, Peng Fei He
2017 arXiv   pre-print
Furthermore, we discuss relations between monadic NM-algebras and some related structures, likeness modal NM-algebras and rough approximation spaces.  ...  Finally, we present monadic NM-logic and prove the (chain) completeness of monadic NM-logic based on monadic NM-algebras.  ...  the same as that of monadic bounded residuated lattice in [20] .  ... 
arXiv:1709.04832v1 fatcat:5a7z5knijvelnhvubi6345btx4

Many-valued logics [chapter]

Erich Peter Klement, Radko Mesiar, Endre Pap
2000 Trends in Logic  
And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics.  ...  The paper considers the fundamental notions of many-valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics.  ...  , which have an additional residuation operation connected with the semigroup operation via an adjointness condition like (28) , and which satisfy a pre-linearity condition like (18) .  ... 
doi:10.1007/978-94-015-9540-7_11 fatcat:zzdkno6gxfam3i2b66quaac5ni
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