Dual digraphs of finite semidistributive lattices

• Autores: Andrew Craig, Miroslav Haviar, José São João
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 24, Nº. 3, 2022, págs. 369-392
• Idioma: inglés
• DOI: 10.56754/0719-0646.2403.0369
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen Se caracterizan los digrafos duales de reticulados finitos unión-semidistributivos, encuentro-semidistributivos y semidistributivos. Los vértices de los digrafos duales son pares filtro-ideales disjuntos maximales del reticulado. El enfoque usado combina las representaciones de reticulados arbitrarios de Urquhart (1978) and Ploščica (1995). Los duales de reticulados finitos son vistos principalmente como digrafos TiRS como fueron presentados y estudiados en Craig-Gouveia-Haviar (2015 y 2022). Cuando sea apropiado, también se emplean los dos cuasi-órdenes de Urquhart en los vértices del digrafo dual. Se introducen los vértices transitivos y se estudia su rol en la teoría de dominación de digrafos. En particular, se caracterizan los reticulados finitos con la propiedad que en sus digrafos TiRS duales los vértices transitivos forman un conjunto dominante (respectivamente un conjunto dominante interior). Se entrega una caracterización de reticulados encuentro- y unión-semidistributivos a través de sistemas de clausura mínima en el conjunto de vértices de sus digrafos duales.

  • English

    Abstract 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 Ploščica (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 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.

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