Nodal Filters in Equality Algebras

Nodal filters provide an order-theoretic approach to the study of the filter structure of algebraic systems. In this paper, we investigate nodal filters in equality algebras, with particular emphasis on nodal filters of types 1, 2, and 3. We first establish several structural properties of these filters and examine their relationships with positive implicative and implicative filters. In particular, we show that, under suitable conditions, such as purity and density, important classes of filters are nodal. We also provide examples and counterexamples to illustrate the scope and limitations of these results. We then introduce operations on the set of all nodal filters of type 1 and study the resulting algebraic structure. We prove that this structure forms an L-algebra and a bounded distributive lattice. Furthermore, by defining a suitable operation, we show that it is a residuated lattice, an MTL-algebra and a BL-algebra. We also establish that the resulting structure is a semi-De Morgan algebra, a bounded equality algebra, and a Stonean equality algebra. Finally, several additional algebraic identities and order-theoretic properties are obtained, providing further insight into the interaction between nodal filters and the algebraic structure induced by their operations.