A topological duality for some lattice ordered algebraic structures including \(\ell\)-groups
The author develops a topological duality for implicative lattices. Implicative lattices are distributive lattices with an additional binary operation which is a dual lattice homomorphism in its first coordinate and a lattice homomorphism in the second coordinate. In this duality the implication corresponds to an order-preserving continuous binary operation. The image of this operation does not lie entirely within the prime filters; it may contain trivial filters. This is remedied by using for the dual space of a distributive lattice the Stone space of the lattice obtained by adding a (possibly new) top and bottom, and then keeping the order to be able to recognize the extra maximum and minimum prime filters that are created. This duality for distributive lattices gives a smooth description of the dual space of an unbounded lattice and allows the author to describe the binary operation obtained from the implication as an operation on the dual space. By specializing his duality for implicative lattices, the author obtains topological dualities for \(l\)-groups and for Abelian \(l\)-groups, since these can be viewed as equational subclasses of implicative lattices with a distinguished element. In the case of Abelian \(l\)-groups, the dual spaces are compact ordered Abelian topological semigroups. Even though these are distributive lattices with additional operation(s), the dualities obtained here are not directly related to the ones obtained by \textit{R. Goldblatt} [Ann. Pure Appl. Logic 44, 173-242 (1989; Zbl 0722.08005)] and other authors who have studied dualities and canonical extensions for distributive lattices with operators and other additional operations. The author obtains a binary operation on the dual space rather than just at ternary relation. And as a consequence, the operation obtained does not share the equational properties of the implication that it came from.
scientific article; zbMATH DE number 638594