Monadic Distributive Lattices

Monadic distributive lattices and monadic augmented Kripke frames

“…On each bounded lattice we can define a special quantifier, namely the indiscrete or simple quantifier given by the prescription ∇0 = 0 and ∇x = 1 for each x ∈ L, x = 0. If L is an m-lattice and ∇ is the simple quantifier, taking into account the results established in [9], we can assert that △1 = 1 and △x = 0 for all x ∈ L, x = 1. In this case, we say that (∇, △) is simple and it will play an important role in the characterization of simple m-lattices.…”

“…Remark 2.2 Let (L, △, ∇) be an m-lattice and P, T filters of L. Then, △ −1 (T ) ⊆ P if and only if T ∩ △(L) ⊆ P . Lemma 2.4 (see [9]) Let L be a distributive lattice, △ an interior operator on L and a ∈ L. If F ⊆ L is a filter such that △a / ∈ F , then there is Q ∈ X(L) satisfying the following conditions:…”

“…Lemma 2.5 (see [9]) Let L be a distributive lattice and △ an interior operator on L. If S, T ∈ X(L) are such that T ∩ △(L) ⊆ S then there is R ∈ X(L) satisfying the following conditions:…”

“…In [9], following the research started in [7], we defined the category mP whose objects are mP-spaces and whose morphisms are mP-functions ([9, p 7]) and we showed that it is dually equivalent to the category M of monadic distributive lattices and their corresponding homomorphisms. This duality allowed us to characterized simple and subdirectly irreducible but not simple algebras in this variety.…”