Dualities for subresiduated lattices (Q2240721)

A subresiduated lattice is determined by a bounded lattice \(L\) together with a bounded sublattice \(D\leq L\) where \(D\) has the property that for every \(a, b\in L\) there is a largest \(c\in D\) such that \(c\wedge a\leq b\). One writes \(a\to b\) for this \(c\). The class of subresiduated lattices is equationally definable in the language \(\{\wedge, \vee, \to , 0, 1\}\), and the variety of subresiduated lattices properly contains the variety of Heyting algebras. (A subresiduated lattice is a Heyting algebra exactly when \(D=L\).) This paper establishes a dual equivalence between the category of subresiduated lattices and the category of \(p\)-spectral spaces. Then, the paper establishes a bitopological style duality between the category of subresiduated lattices and the category subresiduated spaces.