Abstract:It is well known that, in Boolean algebras, the notions of precontact relation, quasi-modal operator, and subordination relation are interdefinable. In contrast, within the setting of distributive lattices, this equivalence holds only between quasi-modal operators and subordination relations, but not with precontact relations. In this paper, we study the class of bounded distributive lattices equipped with a precontact relation, referred to as precontact lattices. We also examine how conditions imposed on the precontact relation correspond to first-order properties in the Priestley dual. In addition, we characterize precontact substructures and strong precontact sublattices of precontact lattices in terms of a suitable lattice preorder. Finally, we introduce a notion of precontact congruence and identify the corresponding closed sets in the dual space.
From: Sergio A. Celani [view email]
[v1]
Tue, 23 Jun 2026 20:05:31 UTC (30 KB)
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