Nodal Filters in Equality Algebras
- 1Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran
- 2Department of Mathematics, Graduate University of Advanced Technology, Kerman 76311-33131, Iran
- 3Center for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia
Abstract
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.
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BibTeX
@article{dehaghi2026nodal,
title = {{Nodal Filters in Equality Algebras}},
author = {Hamid Reza Asadi Dehaghi and Mohammad Mahdi Zahedi and Hashem Bordbar and Ali Iranmanesh},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100766},
url = {https://www.mdpi.com/2075-1680/15/10/766},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Nodal Filters in Equality Algebras
AU - Hamid Reza Asadi Dehaghi
AU - Mohammad Mahdi Zahedi
AU - Hashem Bordbar
AU - Ali Iranmanesh
PY - 2026
DA - 2026/10/13
JO - Axioms
DO - 10.3390/axioms15100766
UR - https://www.mdpi.com/2075-1680/15/10/766
AB - 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.
SN - 2075-1680
ER -