Li-ideals of implicative almost distributive lattices
Alachew Amaneh Mechderso, Tilahun Mekonnen Munie · F1000Research · 2025
Background In this paper, we introduce the concept of H-implicative almost distributive lattices, which are a special class of lattices with both implicative and almost distributive properties. This study is aimed at extending the understanding of lattice structures and their ideals, particularly focusing on LI-ideals, a novel concept within H-implicative almost distributive lattices. Methods We define an LI-ideal of an implicative almost distributive lattice L and investigate its properties. The paper demonstrates that every LI-ideal in L is an almost distributive lattice ideal of L. Additionally, we explore the relationship between filters and LI-ideals, and we study the process of generating an LI-ideal from a given set. Lastly, we examine the construction of quotient structures via LI-ideals. Results We present several examples showing that every almost distributive lattice ideal is also an LI-ideal in an H-implicative almost distributive lattice. The study establishes key relationships between the concepts of filters and LI-ideals. Furthermore, we provide a method for generating an LI-ideal from a set and construct a quotient structure using an LI-ideal. Conclusions The paper introduces new concepts and relationships within the study of H-implicative almost distributive lattices. Our findings demonstrate the interconnection between almost distributive lattice ideals and LI-ideals and offer insights into how these ideals can be generated and used to construct quotient structures. This work provides a deeper understanding of lattice theory and opens new avenues for further research in the area of lattice ideals.