On Semi-Simplicity Results in Residuated Lattices

Esmaeil Rostami

doi:10.18778/0138-0680.2026.24

Authors

Esmaeil Rostami Mahani Math Center, Afzalipour Research Institute, Shahid Bahonar University of Kerman, Kerman, Iran; Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran Author https://orcid.org/0000-0002-8898-4854

DOI:

https://doi.org/10.18778/0138-0680.2026.24

Keywords:

simple filter, semi-simple filter, socle of a residuated lattice, independent family of filters

Abstract

In this paper, we develop the theory of residuated lattices by introducing and studying several new types of filters and related concepts, including semi-simple filters, essential filters, the socle of a filter, and independent families of filters. Our primary goal is to understand the inner structure of residuated lattices by analyzing these new objects. First, we establish the key properties of simple and essential filters. Next, we provide both algebraic and topological characterizations for identifying when a filter is simple or essential. Furthermore, we use the concepts of the socle and independent families of filters to consider deeper into the structure of filters and the residuated lattice itself. Also, several characterizations for semi-simple filters and semi-simple residuated lattices are provided. Using these characterizations, we establish a fundamental logical characterization of semi-simplicity, proving that a residuated lattice is semi-simple if and only if its frame of filters satisfies the law of excluded middle which, in this setting, is equivalent to the logical principles of double negation, material implication, and the contraposition law. A central result shows that for finite residuated lattices, being semi-simple is equivalent to being hyperarchimedean, highlighting the natural connection between these concepts. Complementary results deepen our understanding of the relation between simple and maximal filters in residuated lattices, and a comparison between a ring’s semi-simplicity and the semi-simplicity of the residuated lattice of its ideals are also established.

References

A. Ahadpanah, L. Torkzadeh, Fuzzy Noetherian and Artinian Residuated Lattice, U.P.B. Scientific Bulletin, Series A: Applied Mathematics and Physics, vol. 77(4) (2015), pp. 23–32.

F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, 2nd ed., vol. 13 of Graduate Texts in Mathematics, Springer-Verlag, New York (1992), DOI: https://doi.org/10.1007/978-1-4612-4418-9.

A. Borumand Saeid, M. Pourkhatoun, Obstinate Filters in Residuated Lattices, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 55(4) (2012), pp. 413–422, tome 55(103).

D. Buşneag, D. Piciu, A New Approach for Classification of Filters in Residuated Lattices, Fuzzy Sets and Systems, vol. 260 (2015), pp. 121–130, DOI: https://doi.org/10.1016/j.fss.2014.07.022.

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel, Semisimple Semirings with Respect to Co-Ideals Theory, Asian-European Journal of Mathematics, vol. 11(5) (2018), p. 1850069, DOI: https://doi.org/10.1142/S1793557118500699.

R. Engelking, General Topology, revised and completed ed., vol. 6 of Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin (1989).

N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, 1st ed., vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam (2007).

G. Georgescu, D. Cheptea, C. Mureşan, Algebraic and Topological Results on Lifting Properties in Residuated Lattices, Fuzzy Sets and Systems, vol. 271 (2015), pp. 102–132, DOI: https://doi.org/10.1016/j.fss.2014.11.007.

P. Hájek, Metamathematics of Fuzzy Logic, 1st ed., vol. 4 of Trends in Logic, Kluwer Academic Publishers, Dordrecht (1998), DOI: https://doi.org/10.1007/978-94-011-5300-3, a softcover edition was published in 2001, ISBN 978-1-4020-0370-7.

M. Haveshki, A. Borumand Saeid, E. Eslami, Some Types of Filters in BL Algebras, Soft Computing, vol. 10(8) (2006), pp. 657–664, DOI: https://doi.org/10.1007/s00500-005-0534-4.

U. Hebisch, H. J. Weinert, Semisimple Classes of Semirings, Algebra Colloquium, vol. 9(2) (2002), pp. 177–196.

L.-C. Holdon, New Topology in Residuated Lattices, Open Mathematics, vol. 16(1) (2018), pp. 1104–1127, DOI: https://doi.org/10.1515/math-2018-0092.

L.-C. Holdon, On Ideals in De Morgan Residuated Lattices, Kybernetika, vol. 54(3) (2018), pp. 443–475, DOI: https://doi.org/10.14736/kyb-2018-3-0443.

A. Iorgulescu, Algebras of Logic as BCK Algebras, Editura ASE, Bucharest (2008).

Y. Katsov, T. G. Nam, N. X. Tuyen, On Subtractive Semisimple Semirings,Algebra Colloquium, vol. 16(3) (2009), pp. 415–426, DOI: https://doi.org/10.1142/S1005386709000406.

T.-Y. Lam, A First Course in Noncommutative Rings, 2nd ed., vol. 131 of Graduate Texts in Mathematics, Springer, New York (2001), DOI: https://doi.org/10.1007/978-1-4419-8616-0.

L. Liu, K. Li, Boolean Filters and Positive Implicative Filters of Residuated Lattices, Information Sciences, vol. 177(24) (2007), pp. 5725–5738, DOI: https://doi.org/10.1016/j.ins.2007.07.014.

S. Motamed, J. Moghaderi, Noetherian and Artinian BL-Algebras, Soft Computing, vol. 16(11) (2012), pp. 1989–1994, DOI: https://doi.org/10.1007/s00500-012-0876-7.

D. Piciu, Algebras of Fuzzy Logic, 1st ed., Editura Universitaria, Craiova (2007).

S. V. Tchoffo Foka, M. Tonga, Rings and Residuated Lattices Whose Fuzzy Ideals Form a Boolean Algebra, Soft Computing, vol. 26(2) (2022), pp. 535–539, DOI: https://doi.org/10.1007/s00500-021-06545-z.

E. Turunen, Mathematics Behind Fuzzy Logic, Advances in Soft Computing, Physica-Verlag, Heidelberg (1999).

F. Wang, H. Kim, Foundations of Commutative Rings and Their Modules, 1st ed., vol. 22 of Algebra and Applications, Springer, Singapore (2016), DOI: https://doi.org/10.1007/978-981-10-3337-7.

M. Ward, R. P. Dilworth, Residuated Lattices, Transactions of the American Mathematical Society, vol. 45(3) (1939), pp. 335–354, DOI: https://doi.org/10.1090/S0002-9947-1939-1501995-3, reprinted in K. P. Bogart, R. Freese, and J. P. S. Kung (eds.), The Dilworth Theorems: Selected Papers of Robert P. Dilworth, Birkhäuser, 1990, pp. 317–336.

R. Wisbauer, Foundations of Module and Ring Theory: A Handbook for Study and Research, revised and updated english edition ed., vol. 3 of Algebra, Logic and Applications, Gordon and Breach Science Publishers, Philadelphia (1991), DOI: https://doi.org/10.1201/9780203755532, digitally reissued by Routledge in 2018.