Duality for bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon

According to G. Birkhoff, there is categorical duality between the category of bi-algebraic distributive $(0,1)$-lattices with complete $(0,1)$-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of $(0,1)$-lattices generated by the pentagon, the $5$-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of $(0,1)$-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of $(0,1)$-lattices is either a $2$-element chain or has uncountably many elements and is not distributive.