Equational classes of totally ordered modal lattices (Q1970930)

A modal chain is a bounded totally ordered lattice \(L\) together with a (modal) operator \(j: L\to L\) satisfying \(j(0)= 0\) and \(j(a\vee b)= j(a)\vee j(b)\). Priestley duality for distributive lattices with operators -- as established by Cignoli, Hansoul or the author -- is adapted to the variety CH generated by the modal chains. The dual spaces of the subdirectly irreducible modal chains are characterized. This enables a complete description of the lattice of all subvarieties of CH. Also an equational basis is given for each such subvariety. For instance, CH itself admits the following defining axioms: a) \(j(a\wedge b)= j(a)\wedge j(b)\), b) \((a\wedge j(b))\vee (b\wedge j(a))\leq (a\wedge b)\vee (j(a)\wedge j(b))\).