Study of canonical extensions of BL-algebras

Canonical extensions of lattice ordered algebras provide an algebraic formulation of what is otherwise treated via topological duality or relational methods. They were firstly introduced by Jónsson and Tarski for Boolean algebras with operators (see [8] and [9]) and generalized for distributive lattices with different operations in [5], [4] and [3]. If A = (A, {fi, i ∈ I}) is a distributive lattice with operations, the canonical extension Aσ of the lattice (A,∧,∨) is a doubly algebraic distributive lattice that contains A as separating and compact sublattice. The main problem is to extend the extra operations {fi, i ∈ I} to Aσ and check if this new structure is an algebra in the same class of A. There are two different ways to extend an operation f : one is the canonical extension fσ and the other is the dual canonical extension fπ (see [5]). This gives us two possible candidates for the canonical extension of A, namely the canonical extension Aσ and the dual canonical Aπ. A class of algebras is called σ-canonical or π-canonical if it is closed under canonical or dual canonical extensions respectively. BL-algebras were introduced by Hájek (see [7]) as the algebraic counterpart of basic logic. Since BL-algebras can be viewed as distributive lattices with additional operations, one can analyze the canonicity of different subvarieties of these algebras. BL-algebras form a variety BL that contains as important subvarieties the variety of MV-algebras, the variety of Product algebras and the variety of Gödel algebras (linear Heyting algebras). The study of canonicity in these