Representations and embeddings of M-distributive lattices. (English) Zbl 0551.06014

Bandelt, Hans-J.; Erné, Marcel

Houston J. Math. 10, 315-324 (1984).

Continuous lattices are characterized by the distributivity of arbitrary meets over up-directed joins. The authors consider complete lattices in which arbitrary meets distribute over joins of sets from a certain system M. They discuss the transfer of basic properties of continuous lattices to this generalized context of M-distributive lattices. Questions like the idempotency of the M-below relation or M-embeddings in a cube (i.e. in a power of [0,1]) are considered. Results are applied to the system \(M_ f\) of all finitely generated lower ends. One gets the result of S. Papert [Proc. Camb. Philos. Soc. 55, 172-176 (1959; Zbl 0178.337)] that \(M_ f\)-distributive lattices having the idempotent \(M_ f\)- below relation coincide with lattices of closed sets of a topological space. One should relate \(M_ f\)-distributive lattices to frames (in the sense of P. T. Johnstone [Stone spaces (1982; Zbl 0499.54001)]) because it is easy to see that any \(M_ f\)-distributive lattice is the dual of a frame.

Reviewer: J.Rosický

MSC:

06D10	Complete distributivity
06B23	Complete lattices, completions
06D05	Structure and representation theory of distributive lattices

Keywords:

complete lattices; continuous lattices; M-distributive lattices; finitely generated lower ends; lattices of closed sets; frames

Citations:

Zbl 0178.337; Zbl 0499.54001