×

Categories of algebraic contexts equivalent to idempotent semirings and domain semirings. (English) Zbl 1364.68337

Kahl, Wolfram (ed.) et al., Relational and algebraic methods in computer science. 13th international conference, RAMiCS 2012, Cambridge, UK, September 17–20, 2012. Proceedings. Berlin: Springer (ISBN 978-3-642-33313-2/pbk). Lecture Notes in Computer Science 7560, 195-206 (2012).
Summary: A categorical equivalence between algebraic contexts with relational morphisms and join-semilattices with homomorphisms is presented and extended to idempotent semirings and domain semirings. These contexts are the Kripke structures for idempotent semirings and allow more efficient computations on finite models because they can be logarithmically smaller than the original semiring. Some examples and constructions such as matrix semirings are also considered.
For the entire collection see [Zbl 1246.68043].

MSC:

68T30 Knowledge representation
06A12 Semilattices
16Y60 Semirings
18A23 Natural morphisms, dinatural morphisms
18B99 Special categories
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birkhoff, G.: Lattice Theory, 3rd edn. AMS Colloquium Publications, vol. XXV. American Mathematical Society, Providence (1967) · Zbl 0153.02501
[2] Davey, B., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002) · Zbl 1002.06001
[3] Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. Journal of Symbolic Logic 70(3), 713–740 (2005) · Zbl 1101.03021
[4] Erné, M.: Categories of contexts (preprint), http://www.iazd.uni-hannover.de/ erne/preprints/CatConts.pdf
[5] Galatos, N., Jipsen, P.: Residuated frames with applications to decidability. To appear in Transactions of the American Math. Soc. · Zbl 1285.03077
[6] Ganter, B., Wille, R.: Formal concept analysis. Mathematical foundations. Springer, Berlin (1999) · Zbl 0909.06001
[7] Gehrke, M.: Generalized Kripke frames. Studia Logica 84, 241–275 (2006) · Zbl 1115.03013
[8] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press (2003) · Zbl 1088.06001
[9] Hartung, G.: A topological representation of lattices. Algebra Universalis 29(2), 273–299 (1992) · Zbl 0790.06005
[10] Hitzler, P., Krötzsch, M., Zhang, G.-Q.: A categorical view on algebraic lattices in formal concept analysis. Fundamenta Informaticae 74, 301–328 (2006) · Zbl 1104.06002
[11] Hofmann, K.H., Mislove, M.W., Stralka, A.R.: The Pontryagin Duality of Compact 0-Dimensional Semilattices and Its Applications. Lecture Notes in Mathematics, vol. 396. Springer (1974) · Zbl 0281.06004
[12] Moshier, M.A.: A relational category of formal contexts (preprint)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.