Algebras for Galois-style connections and their discrete duality

Bounded distributive lattices with pairs of operators forming Galois, dual Galois, residuation and dual residuation connections, respectively, are considered. A discrete duality for these classes of algebras and the corresponding classes of relational systems is established. These results are extended to the fuzzy logic MTL.

[1]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[2]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[3]  Jouni Järvinen.  Pawlak's Information Systems in Terms of Galois Connections and Functional Dependencies , 2007, Fundam. Informaticae.

[4]  Ewa Orlowska,et al.  Relational Representation Theorems for General Lattices with Negations , 2006, RelMiCS.

[5]  Anna Maria Radzikowska,et al.  Fuzzy modal-like approximation operators based on double residuated lattices , 2006, J. Appl. Non Class. Logics.

[6]  Sergio A. Celani,et al.  Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras , 2006 .

[7]  O. V. Zeevald.  Fuzzy logics with modalities , 2006 .

[8]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[9]  Marcel Erné,et al.  A Primer on Galois Connections , 1993 .

[10]  Ivo Düntsch,et al.  Relational Semantics Through Duality , 2005, RelMiCS.

[11]  Anna Maria Radzikowska,et al.  Fuzzy Information Relations and Operators: An Algebraic Approach Based on Residuated Lattices , 2006, Theory and Applications of Relational Structures as Knowledge Instruments.

[12]  Ewa Orlowska,et al.  Relational Methods for Computer Science Applications , 2001 .

[13]  M. Stone.  Topological representations of distributive lattices and Brouwerian logics , 1938 .

[14]  Ewa Orlowska,et al.  Context Algebras, Context Frames, and Their Discrete Duality , 2008, Trans. Rough Sets.

[15]  Anna Maria Radzikowska,et al.  Characterisation of main classes of fuzzy relations using fuzzy modal operators , 2005, Fuzzy Sets Syst..

[16]  Stéphane Demri,et al.  Structures of Information , 2002 .

[17]  Gary M. Hardegree,et al.  Algebraic Methods in Philosophical Logic , 2001 .

[18]  Ivo Düntsch,et al.  Modal-style operators in qualitative data analysis , 2002, 2002 IEEE International Conference on Data Mining, 2002. Proceedings..

[19]  Ewa Orlowska,et al.  Relational Representation Theorems for Lattices with Negations: A Survey , 2006, Theory and Applications of Relational Structures as Knowledge Instruments.

[20]  Alasdair Urquhart,et al.  Duality for algebras of relevant logics , 1996, Stud Logica.

[21]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[22]  Ivo Düntsch,et al.  Beyond modalities: sufficiency and mixed algebras , 2001 .

[23]  Ewa Orlowska,et al.  Discrete Dualities for Heyting Algebras with Operators , 2007, Fundam. Informaticae.

[24]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[25]  Ewa Orlowska,et al.  Duality via Truth: Semantic frameworks for lattice-based logics , 2005, Log. J. IGPL.

[26]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[27]  Ewa Orlowska,et al.  Discrete Duality and Its Applications to Reasoning with Incomplete Information , 2007, RSEISP.

[28]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[29]  Ivo Düntsch,et al.  Boolean algebras arising from information systems , 2004, Ann. Pure Appl. Log..

[30]  Stéphane Demri,et al.  Incomplete Information: Structure, Inference, Complexity , 2002, Monographs in Theoretical Computer Science An EATCS Series.

[31]  Ewa Orłowska,et al.  Dual Tableaux: Foundations, Methodology, Case Studies , 2010 .

[32]  Ivo Düntsch,et al.  A discrete duality between apartness algebras and apartness frames , 2008, J. Appl. Non Class. Logics.