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Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices

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Abstract

We extend Cayley’s and Holland’s representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integral residuated lattices and many of its subvarieties.

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Notes

  1. Complete residuated lattices where multiplication coincides with meet give rise to locales, examples of which are formed by considering the open sets of a topological space. If A is the powerset of a topological space X, viewed as a residuated lattice, then the topological interior operator σ on X is a conucleus and A σ is the locale of open sets of X. Note that if S⊆A is a basis for X, then σ S gives the associated interior operator for the topology.

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Acknowledgements

The authors would like to thank the anonymous referees for their useful suggestions. The work of the second author was partly supported by the grant P202/11/1632 of the Czech Science Foundation and partly by the Institutional Research Plan AV0Z10300504.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, 2360 S. Gaylord St., Denver, CO, 80208, USA

    Nikolaos Galatos

  2. Institute of Computer Sciences, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 271/2, 182 07, Prague 8, Czech Republic

    Rostislav Horčík

Authors
  1. Nikolaos Galatos
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  2. Rostislav Horčík
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Correspondence to Nikolaos Galatos.

Additional information

Communicated by Jimmie D. Lawson.

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Galatos, N., Horčík, R. Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices. Semigroup Forum 87, 569–589 (2013). https://doi.org/10.1007/s00233-013-9513-8

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  • DOI: https://doi.org/10.1007/s00233-013-9513-8

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