Morphisms in Context

Morphisms constitute a general tool for modelling complex relationships between mathematical objects in a disciplined fashion. In Formal Concept Analysis (FCA), morphisms can be used for the study of structural properties of knowledge represented in formal contexts, with applications to data transformation and merging. In this paper we present a comprehensive treatment of some of the most important morphisms in FCA and their relationships, including dual bonds, scale measures, infomorphisms, and their respective relations to Galois connections. We summarize our results in a concept lattice that cumulates the relationships among the considered morphisms. The purpose of this work is to lay a foundation for applications of FCA in ontology research and similar areas, where morphisms help formalize the interplay among distributed knowledge bases.

[1]  Gunther Schmidt,et al.  Relational Methods in Computer Science , 1999, Inf. Sci..

[2]  F. W. Lawvere,et al.  Sets for Mathematics , 2003 .

[3]  Marc Ehrig,et al.  What is Ontology Merging? - A Category-Theoretical Perspective using Pushouts , 2005, AAAI 2005.

[4]  Bernhard Ganter,et al.  Relational Galois Connections , 2007, ICFCA.

[5]  Joseph A. Goguen.  Three Perspectives on Information Integration , 2005, Semantic Interoperability and Integration.

[6]  Pascal Hitzler,et al.  A Categorical View on Algebraic Lattices in Formal Concept Analysis , 2004, Fundam. Informaticae.

[7]  Robert E. Kent.  Distributed Conceptual Structures , 2001, RelMiCS.

[8]  H.C.M. de Swart,et al.  Relational methods in computer science : 6th International Conference, RelMiCS 2001 and 1st Workshop of COST Action 274 TARSKI, Oisterwijk, The Netherlands, October 16-21, 2001 : revised papers , 2002 .

[9]  Vaughan Pratt,et al.  Chu spaces as a semantic bridge between linear logic and mathematics , 2003, Theor. Comput. Sci..

[10]  Markus Krtzsch,et al.  Morphisms in Logic, Topology, and Formal Concept Analysis , 2005 .

[11]  Jon Barwise,et al.  Information Flow: The Logic of Distributed Systems , 1997 .

[12]  Joseph A. Goguen,et al.  Institutions: abstract model theory for specification and programming , 1992, JACM.

[13]  Amit P. Sheth,et al.  Semantic Interoperability and Integration , 2005, Semantic Interoperability and Integration.

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

[15]  Robert E. Kent.  Semantic Integration in the Information Flow Framework , 2005, Semantic Interoperability and Integration.