Coherent Filters of Pseudocomplemented 1-Distributive Lattices

This work explores coherent filters in the framework of pseudocomplemented 1-distributive lattices. After reviewing the basic properties of such lattices and their pseudocomplements, we introduce the notion of coherent filters and establish conditions under which a filter is coherent. The study further examines the relationships between coherent, strongly coherent, and τ-closed filters, showing how these concepts interact with classical structures such as p-filters and D-filters. Several equivalent characterizations are derived, linking coherence with closure, pseudocomplements, and annihilators. In addition, we investigate semi Stone and Stone lattices, proving that a pseudocomplemented 1-distributive lattice is semi Stone precisely when every τ-closed filter is strongly coherent. This provides a new structural perspective on the role of coherence in lattice theory. By generalizing results previously known in distributive lattices, the paper offers a unified approach to understanding filter behavior in broader algebraic settings, with potential implications for further developments in lattice theory and related algebraic systems.