Generalizations of Boolean products for lattice-ordered algebras
Topological dualities have been very effective tools for various classes of algebras, such as Boolean algebras with Boolean spaces as duals, distributive lattices with Priestley spaces as duals, and Heyting algebras with Esakia spaces as duals. Boolean spaces have also been applied to the representation of algebras by Boolean powers and (weak) Boolean products, where the latter are also known as algebras of global sections of sheaves of algebras over Boolean spaces. In this paper, generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, the author shows that \(\text{FL}_{\omega}\)-algebras decompose as a poset product over any finite set of join-irreducible strongly central elements, and that bounded \(n\)-potent GBL-algebras are represented as Esakia products of simple \(n\)-potent MV-algebras.
scientific article; zbMATH DE number 5624911