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Boolean and ortho fuzzy subset logics [article]

Daniel J. Greenhoe
2015 arXiv   pre-print
However, this paper examines these three results in detail, and shows that "hidden" in the hypotheses of the three is the assumption that the operator pair (meet,join) is pointwise evaluated.  ...  idempotency, non-contradiction, excluded middle, and distributivity.  ...  APPENDIX B introduces several types of negation, describes some of their properties, and shows where fuzzy negation, ortho negation, and Boolean negation "fit" into the larger structure of negations in  ... 
arXiv:1409.4222v2 fatcat:xb6g4svrrncuploqlmzz7ct2nq

Page 1692 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
Liu shows that a set-pair algebra is a soft algebra, that is, a distributive lattice with a reversing involution and also a fuzzy lattice (i.e. completely distributive lattice).  ...  Moreover, a representation theorem for soft algebras in terms of set-pair algebras is established. Ying Ming Liu (Chengdu) 2000c:06024 06D30 06B10 Liu, Zheng Xin; Jia, Wu M -ideals of fuzzy lattices.  ... 

Extended-order algebras and fuzzy implicators

M. E. Della Stella, C. Guido
2012 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
In this work we reconsider the notion of implicator in a complete lattice L and discuss its properties, taking into account the viewpoint of the implication operation of classes of (weak) extended-order  ...  In fact, such an implication, that is an extension of an order relation, can be viewed as an implicator in L, whose properties depend on those characterizing the structure of the algebra.  ...  The authors are grateful to the referees for their useful comments on the first version of this paper.  ... 
doi:10.1007/s00500-012-0840-6 fatcat:gg2vnx5qjjhcpp43sdr563gsj4

Monadic Bounded Residuated Lattices

Jiří Rachůnek, Dana Šalounová
2011 Order  
In the paper we introduce and investigate monadic bounded integral residuated lattices which can be taken as a generalization of algebraic models of the predicate calculi of those logics in which only  ...  Bounded integral residuated lattices form a large class of algebras which contains algebraic counterparts of several propositional logics behind many-valued reasoning and intuitionistic logic.  ...  (= weak pseudoBL-algebras) [13, 16] , i.e. algebras of the monoidal t-norm based logic MT L [12], the logic of left-continuous t-norms, and of its non-commutative version PMT L [21].  ... 
doi:10.1007/s11083-011-9236-y fatcat:x4woubxx7favnjjpgoy54n763i

Algebraic Structure of Adjoint Triples Generating a Weak Negation on the Unit Interval [chapter]

M. Eugenia Cornejo, Jesús Medina, Eloísa Ramírez-Poussa
2020 Lecture Notes in Computer Science  
This paper presents a theoretical research about the relationship between weak negations and adjoint negations.  ...  Adjoint negations are a generalization of residuated negations built from the implications of an adjoint triple.  ...  , Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124.  ... 
doi:10.1007/978-3-030-52705-1_25 fatcat:gzcakcjkpvblbecfasija5sgui

Linking Mathematical Morphology and L-Fuzzy Concepts

Cristina Alcalde, Ana Burusco, Humberto Bustince, Ramon Fuentes-Gonzalez, Mikel Sesma-Sara
2017 International Journal of Uncertainty Fuzziness and Knowledge-Based Systems  
Given an L-fuzzy implication funcion I, it can be used to defined a conjunctive operator C (i.e., a commutative, associative and increasing operator) as follows. Definition 3.  ...  In this paper we study the relation between L-fuzzy morphology and L-fuzzy concepts over complete lattices.  ...  Acknowledgements This work has been partially supported by the Research Group "Intelligent Systems and Energy (SI+E)" of the UPV/EHU, under Grant GIU 16/54, and by project TIN2016-77356-P(AEI/FEDER, UE  ... 
doi:10.1142/s0218488517400049 fatcat:ciek34qo7jg4dju4oourccgto4

Fixpointed Idempotent Uninorm (Based) Logics

Eunsuk Yang
2019 Mathematics  
The algebraic structures corresponding to the systems are then defined, and the results of the associated algebraic completeness are provided.  ...  Idempotent uninorms are simply defined by fixpointed negations.  ...  Conflicts of Interest: The author declares no conflict of interest.  ... 
doi:10.3390/math7010107 fatcat:dh5v7om32ffkraiugwyliuvhue

Generalized qualitative Sugeno integrals

Didier Dubois, Henri Prade, Agnès Rico, Bruno Teheux
2017 Information Sciences  
These fuzzy conjunction and implication connectives are related via a so-called semiduality property, involving an involutive negation.  ...  Sugeno integrals correspond to the case when the fuzzy conjunction is the minimum and the fuzzy implication is Kleene-Dienes implication, in which case integrals and cointegrals coincide.  ...  , and non-commutative fuzzy conjunctions obtained via an involutive negation.  ... 
doi:10.1016/j.ins.2017.05.037 fatcat:if7qlwo3mvfjtla6yksm35icay

Algebraic hierarchy of logics unifying fuzzy logic and quantum logic [article]

Andreas de Vries
2007 arXiv   pre-print
New examples of a non-contradictory logic violating the conjunctive De Morgan law, and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given.  ...  law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other.  ...  An essential step to this unifying picture is the notion of a general fuzzy negation holding the conditions of weak double negation and antitony, as well as the Boolean boundary condition.  ... 
arXiv:0707.2161v1 fatcat:un3npcgoujcplpur72urncmkaq

Graded cubes of opposition and possibility theory with fuzzy events

Didier Dubois, Henri Prade, Agnès Rico
2017 International Journal of Approximate Reasoning  
and necessity of fuzzy events in the setting of Choquet integrals, e.g.  ...  Keywords: Possibility theory Fuzzy event Cube of opposition The paper discusses graded extensions of the cube of opposition, a structure that naturally emerges from the square of opposition in philosophical  ...  In this way, we obtained two pairs of implication and conjunction connectives related via semi-duality, due to the involutive negation, each pair being related to the other pair via residuation.  ... 
doi:10.1016/j.ijar.2017.02.006 fatcat:m6dknsrkpnabhhzvgigik7htjy

MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing [article]

Daniel J. Greenhoe
2014 arXiv   pre-print
Used recursively, a sequence of Boolean lattices with decreasing order is generated---a structure that is similar to an MRA.  ...  Examples of symbolic sequences include sequences of logic values, sequences of probabilistic events, and genomic sequences (as used in "genomic signal processing").  ...  ⟂ (Definition 3.2 page 34) ⟹ (¬¬ ) ≤ ¬ by antitone property (Definition 1.72 page 20) ⟹ ≤ ¬ by weak double negation property of negation (Definition 2.13 page 28) ⟹ ⟂ by definition of ⟂ (Definition  ... 
arXiv:1410.4830v1 fatcat:rhbat3udcnc4da5qo5hyhyqzcu

Multi-adjoint algebras versus non-commutative residuated structures

M. Eugenia Cornejo, Jesús Medina, Eloisa Ramírez-Poussa
2015 International Journal of Approximate Reasoning  
Furthermore, a comparison between adjoint negations and the negations introduced in extended-order algebras is presented.  ...  Adjoint triples are an interesting generalization of t-norms and their residuated implications, since they increase the flexibility in the framework where they are used.  ...  Let P be a non-empty set, → : P × P → P a binary operation and ⊤ a fixed element of P.  ... 
doi:10.1016/j.ijar.2015.08.003 fatcat:q27ko7up75an7g3fjnxn4oogfi

Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Application to decision making

Edurne Barrenechea, Javier Fernandez, Miguel Pagola, Francisco Chiclana, Humberto Bustince
2014 Knowledge-Based Systems  
This paper presents a method to construct an interval-valued fuzzy set from a fuzzy set and the representation of the lack of knowledge or ignorance that experts are subject to when they define the membership  ...  values of the elements to that fuzzy set.  ...  Acknowledgment This work has been partially supported by the Research Services of the Universidad Publica de Navarra, and the National Science Foundation of Spain, reference TIN2010-15055.  ... 
doi:10.1016/j.knosys.2013.10.002 fatcat:cye2fwzrdbfqhkpyi4h2w6r5ne

Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

Carles Noguera, Francesc Esteva, Lluís Godo
2010 Information Sciences  
Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains.  ...  We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains.  ...  Acknowledgments We want to thank Vicenzo Marra for his helpful comments on group theory and the anonymous referees for their many comments and remarks that helped us improving much the first  ... 
doi:10.1016/j.ins.2009.12.011 fatcat:gt2fmpoj7jgcxnhot7d4asmwee

A map of dependencies among three-valued logics

D. Ciucci, D. Dubois
2013 Information Sciences  
However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones.  ...  They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested.  ...  As fuzzy logic underlies fuzzy sets, three-valued logics lead to three-valued sets, that can be modeled by pairs of nested sets or orthopairs of disjoint sets [Cattaneo and Ciucci(2003) , Yao(2012) ]  ... 
doi:10.1016/j.ins.2013.06.040 fatcat:ty7uhcjmbvbjrbgrflnz4ww7sa
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