TY - JOUR AU - Craig, Andrew AU - Haviar, Miroslav AU - São João, José PY - 2022/12/21 Y2 - 2024/03/28 TI - Dual digraphs of finite semidistributive lattices JF - CUBO, A Mathematical Journal JA - CUBO VL - 24 IS - 3 SE - Articles DO - 10.56754/0719-0646.2403.0369 UR - https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3205 SP - 369–392 AB - <p class="p1">Dual digraphs of finite join-semidistributive lattices, meet-semidistributive lattices and semidistributive lattices are characterised. The vertices of the dual digraphs are maximal disjoint filter-ideal pairs of the lattice. The approach used here combines representations of arbitrary lattices due to Urquhart (1978) and PlošÄica (1995). The duals of finite lattices are mainly viewed as TiRS digraphs as they were presented and studied in Craig--Gouveia--Haviar (2015 and 2022). When appropriate, Urquhart's two quasi-orders on the vertices of the dual digraph are also employed. Transitive vertices are introduced and their role in the<span class="Apple-converted-space">&nbsp; </span>domination theory of the digraphs is studied. In particular, finite lattices with the property that in their dual TiRS digraphs the transitive vertices form a dominating set (respectively, an in-dominating set) are characterised. A characterisation of both finite meet-and join-semidistributive lattices is provided via minimal closure systems on the set of vertices of their dual digraphs.</p> ER -