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Dualities for subresiduated lattices


A subresiduated lattice is a pair (AD), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\), \(d\wedge a\le b\) if and only if \(d\le c\). This c is denoted by \(a\rightarrow b\). This pair can be regarded as an algebra \(\left<A,\wedge ,\vee ,\rightarrow ,0,1\right>\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\). The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.

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The authors acknowledge many helpful comments from the anonymous referee, which considerably improved the presentation of this paper. The second author was supported by a doctoral fellowship from Comisión de Investigaciones Científicas (CIC-Argentina).

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Correspondence to Hernán J. San Martín.

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Celani, S.A., Nagy, A.L. & Martín, H.J.S. Dualities for subresiduated lattices. Algebra Univers. 82, 59 (2021).

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  • Distributive lattices
  • Subresiduated lattices
  • Spectral duality
  • Bitopological duality

Mathematics Subject Classification

  • 06D05
  • 06D50
  • 03G10