Abstract.
While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates.
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The second and third authors were supported by ARC Discovery Project Grants DP0342459 and DP0556248, respectively.
Received August 31, 2005; accepted in final form March 23,2007.
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Davey, B.A., Jackson, M., Pitkethly, J.G. et al. Natural dualities for semilattice-based algebras. Algebra univers. 57, 463–490 (2007). https://doi.org/10.1007/s00012-007-2061-x
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DOI: https://doi.org/10.1007/s00012-007-2061-x