Topological Insights into Prime and Minimal Prime Filters on Paradistributive Latticoids

Abstract

In this paper, we investigate prime filters and minimal prime filters on a paradistributive latticoid (PDL) and establish significant theoretical results. We demonstrate that the annihilator filter S∙S^\bullet is equal to the intersection of all prime filters not containing SS, providing a novel perspective on filter structures in PDLs. Furthermore, we explore minimal prime filters, analyzing their unique properties and relationships within the PDL framework. Our study also presents equivalent conditions for a PDL to be relatively complemented, offering insights into its algebraic structure. Additionally, we derive and discuss the topological properties of the spaces of prime filters and minimal prime filters, emphasizing their interactions and contributions to the broader understanding of PDLs. These results expand the foundational knowledge of PDLs and provide new tools for analyzing their algebraic and topological characteristics.