Dual spaces of lattices and semidistributive lattices

Abstract

Birkhoff's 1937 dual representation of finite distributive lattices via finite posets was in 1970 extended to a dual representation of arbitrary distributive lattices via compact totally order-disconnected topological spaces by Priestley. This result enabled the development of natural duality theory in the 1980s by Davey and Werner, later on also in collaboration with Clark and Priestley. In 1978 Urquhart extended Priestley's representation to general lattices via compact doubly quasi-ordered topological spaces (L-spaces). In 1995 Ploščica presented Urquhart's representation in the spirit of natural duality theory by replacing, on the dual side, Urquhart's two quasiorders with a digraph relation generalising Priestley's order relation. In this paper we translate, following the spirit of natural duality theory, Urquhart's L-spaces into newly introduced \emph{Ploščica spaces}. We then prove that every Ploščica space is the dual space of some general lattice. Based on the authors' 2022 characterisation of finite join and meet semidistributive lattices via their dual digraphs, we initiate a study of general (possibly infinite) join and meet semidistributive lattices via their dual digraphs. We illustrate our results on examples and formulate three open problems.

Figures (6)

• Figure 1: Examples of finite (non-distributive) lattices and their dual digraphs. For clarity, we omit the loops whose existence follows from the reflexivity of $E$.
• Figure 2: Illustrating the proof of Theorem \ref{['prop:SJimpliesJSD']}
• Figure 3: The infinite semidistributive lattice $\mathbf{O}_\omega$, and the core of its dual space
• Figure 4: The infinite meet but not join semidistributive lattice $\hat{\mathbf{O}}_\omega$, and the core of its dual space
• Figure 5: The infinite lattice $\mathbf{R}$, which is neither join semidistributive nor meet semidistributive, and its dual space.

Theorems & Definitions (40)

• Definition 2.1: U78
• Definition 2.2: P3
• Lemma 2.3
• proof
• Lemma 2.4: cf. Plos95
• Proposition 2.5: Plos95
• Lemma 2.6
• proof
• Lemma 2.7
• proof