Coherent Filters of Pseudocomplemented 1-Distributive Lattices

Chandrani Nag; Syed Md Omar Faruk

doi:doi:10.11648/j.ml.20251103.11

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Coherent Filters of Pseudocomplemented 1-Distributive Lattices

Chandrani Nag

, Syed Md Omar Faruk^*

Published in Mathematics Letters (Volume 11, Issue 3)

Received: 25 August 2025 Accepted: 3 September 2025 Published: 25 September 2025

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Abstract

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.

Published in	Mathematics Letters (Volume 11, Issue 3)
DOI	10.11648/j.ml.20251103.11
Page(s)	60-65
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

1-Distributive Lattice, Pseudocomplemented Lattice, Ideal, Filter, Coherent Filter

References

[1]	T. S. Blyth, Ideals and Filters of Pseudocomplemented Semilattices, Proc. Edinburgh Math. Soc., 23 (1980), 301-316. https://doi.org/10.1017/S0013091500003850
[2]	J. C. Varlet, A Generalization of the Notion of Pseudocomplementedness, Bull. Soc. Sci. Liège, 36 (1967), 149-158.
[3]	G. Grätzer, Lattice Theory: First Concepts and Distributive Lattices, W. H. Freeman, San Francisco, 1971.
[4]	G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1998.
[5]	W. H. Cornish, Congruences on Distributive PseudocomplementedLattices, Bull. Austral. Math. Soc., 82 (1973), 161-179. https://doi.org/10.1017/S0004972700042404
[6]	T. Katriňák, P-Algebras, in Contributions to Lattice Theory, Colloq. Math. Soc. János Bolyai, Vol. 33, Szeged, 1980, pp. 549-573.
[7]	T. Katriˇ nák and P. Mederly, Construction of P-Algebras, Algebra Universalis, 17 (1983), 288-316.
[8]	C. Nag, S. N. Begum, and M. R. Talukder, Some Characterizations of Subclasses of P-Algebras, Southeast Asian Bull. Math., 41 (2017), 535-546.
[9]	C. Nag, S. N. Begum, and M. R. Talukder, P-Ideals and P-Filters of a P-Algebra, Southeast Asian Bull. Math., 42 (2018), 411-424.
[10]	C. Nag, S. N. Begum, and M. R. Talukder, Kernel Ideals and Cokernel Filters of a P-Algebra, Acta Math. Hungar., 154(2) (2018), 279-288. https://doi.org/10.1007/s10474-018-0793-z
[11]	M. S. Rao, Median Filters of Pseudocomplemented Distributive Lattices, Discuss. Math. Gen. Algebra Appl., 44 (2024), 147-161. https://doi.org/10.7151/dmgaa.1446

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APA Style

Nag, C., Faruk, S. M. O. (2025). Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Mathematics Letters, 11(3), 60-65. https://doi.org/10.11648/j.ml.20251103.11

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ACS Style

Nag, C.; Faruk, S. M. O. Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Math. Lett. 2025, 11(3), 60-65. doi: 10.11648/j.ml.20251103.11

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AMA Style

Nag C, Faruk SMO. Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Math Lett. 2025;11(3):60-65. doi: 10.11648/j.ml.20251103.11

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@article{10.11648/j.ml.20251103.11,
  author = {Chandrani Nag and Syed Md Omar Faruk},
  title = {Coherent Filters of Pseudocomplemented 1-Distributive Lattices
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {3},
  pages = {60-65},
  doi = {10.11648/j.ml.20251103.11},
  url = {https://doi.org/10.11648/j.ml.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251103.11},
  abstract = {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.
},
 year = {2025}
}

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TY  - JOUR
T1  - Coherent Filters of Pseudocomplemented 1-Distributive Lattices

AU  - Chandrani Nag
AU  - Syed Md Omar Faruk
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251103.11
DO  - 10.11648/j.ml.20251103.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 60
EP  - 65
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20251103.11
AB  - 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.

VL  - 11
IS  - 3
ER  -

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Author Information

Chandrani Nag

Department of Mathematics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

Contact Email

http://orcid.org/0009-0008-4845-6222
Syed Md Omar Faruk

Department of Mathematics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

Contact Email

http://orcid.org/0009-0004-0591-6648

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Plain Text BibTeX RIS

APA Style

Nag, C., Faruk, S. M. O. (2025). Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Mathematics Letters, 11(3), 60-65. https://doi.org/10.11648/j.ml.20251103.11

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ACS Style

Nag, C.; Faruk, S. M. O. Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Math. Lett. 2025, 11(3), 60-65. doi: 10.11648/j.ml.20251103.11

Copy | Download

AMA Style

Nag C, Faruk SMO. Coherent Filters of Pseudocomplemented 1-Distributive Lattices. Math Lett. 2025;11(3):60-65. doi: 10.11648/j.ml.20251103.11

Copy | Download

@article{10.11648/j.ml.20251103.11,
  author = {Chandrani Nag and Syed Md Omar Faruk},
  title = {Coherent Filters of Pseudocomplemented 1-Distributive Lattices
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {3},
  pages = {60-65},
  doi = {10.11648/j.ml.20251103.11},
  url = {https://doi.org/10.11648/j.ml.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251103.11},
  abstract = {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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Coherent Filters of Pseudocomplemented 1-Distributive Lattices

AU  - Chandrani Nag
AU  - Syed Md Omar Faruk
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251103.11
DO  - 10.11648/j.ml.20251103.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 60
EP  - 65
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20251103.11
AB  - 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.

VL  - 11
IS  - 3
ER  -

Copy | Download