Order DOI 10.1007/s11083-017-9435-2 A Spectral-style Duality for Distributive Posets Luciano J. Gonz´ alez 1 · Ramon Jansana 2 Received: 19 July 2016 / Accepted: 28 June 2017 © Springer Science+Business Media B.V. 2017 Abstract In this paper, we present a topological duality for a category of partially ordered sets that satisfy a distributivity condition studied by David and Ern´ e. We call these posets mo-distributive. Our duality extends a duality given by David and Ern´ e because our category of spaces has the same objects as theirs but the class of morphisms that we consider strictly includes their morphisms. As a consequence of our duality, the duality of David and Ern´ e easily follows. Using the dual spaces of the mo-distributive posets we prove the existence of a particular  1 -completion for mo-distributive posets that might be different from the canonical extension. This allows us to show that the canonical extension of a distributive meet-semilattice is a completely distributive algebraic lattice. Keywords Posets · Distributivity · Topological duality · Completion 1 Introduction The topological dualities for classes of algebras associated with logics arose mainly with M.H. Stone’s work [21] in the mid-1930s when he developed a duality between Boolean algebras and the class of compact, Hausdorff, and zero-dimensional topological spaces, later known as Stone spaces. In the subsequent paper [22], Stone generalizes the previous duality for Boolean algebras to show that the category of bounded distributive lattices and  Luciano J. Gonz´ alez lucianogonzalez@exactas.unlpam.edu.ar Ramon Jansana jansana@ub.edu 1 Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Uruguay 151, 6300 Santa Rosa, Argentina 2 Department de Filosofia, IMUB and BGSMath, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Spain