Categories and General Algebraic Structures with Applications

Categories and General Algebraic Structures with Applications

Generating EQ-algebras from equality algebras with $L$-condition

Articles in Press

Document Type : Research Paper

Authors
Zhe Jie Duan 1
Xiao Long L Xin 2
Rajab Ali Borzooei 3
1 School of Mathematics and Statistics, Weinan Normal University, Weinan,714000, China
2 Bussiness School, Xi'an International University, Xi'an 710048, China
3 Soft Computing Center, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran
10.48308/cgasa.2026.244287.1622
Abstract
Inspired by W. Rump’s foundational work on $L$-algebras and self‑similarity, we introduce a new class of algebras called $L$-equality algebras by adding an implicative condition to equality algebras. Within this framework, we define self‑similar $L$-equality algebras and show that they are equivalent to divisible residuated $EQ$‑algebras satisfying a certain condition, via a naturally induced product operation. We induce filters as $L$-algebras, that is $L$-filters, and filters as equality-algebras, that is $e$-filters, and their prime versions. We verify $L$-filters and $e$-filters coincide, but prime $L$-filters and prime $e$-filters are not equivalent. To establish the reticulations from a bounded L-equality algebra $E$ to a bounded lattice $L$, we first induce and study some new operations $\oplus$ and $\otimes$ on $E$, by which we introduce the concept of reticulations in an axiomatic manner. By utilizing the prime $L$-filters on $E$, we establish the prime $L$-spectrum of $E$, and by using the reticulation, we illustrate that the prime $L$-spectrum of $E$ is homeomorphic to spectrum of $L$. Finally, we discuss the relationship between reticulation on $L$-equality algebras and reticulation on $EQ$-algebras, where the concept of reticulation on $EQ$-algebras is introduced by [X. Zhang, The spectra and reticulation of $EQ$-algebras, Soft Computing]. By utilizing the relationship between $L$-equality algebras and $EQ$-algebras as presented in our article, we conclude that these reticulations are consistent with each other.
Keywords
$L$-equality algebra
$EQ$-algebra
prime $L$-filter
prime $e$-filter
reticulation
spectral space
Subjects
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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 11 September 2026