EXPLORING THE IDEAL LATTICE OF AT-MOST UNIQUELY COMPLEMENTED LATTICES - Dubey, Seema; Kaushal, Namrata (ShodhKosh: Journal of Visual and Performing Arts, 2024)
This paper presents an investigation into the ideal lattice of at-most uniquely complemented lattices, emphasizing their structure and inherent properties. It is shown that a bounded lattice naturally admits a special embedding, and such embedding are always regular in nature. We further establish that the corresponding ideal lattice preserves distributive behavior and identify the precise conditions under which certain ideals attain join-irreducibility. The study also examines complemented elements in relation to principal ideals, distinguishing between bounded and unbounded cases of such lattices. As a significant consequence, we prove that a lattice LLL with at most one complement can be embedded into a kkk-complete uniquely complemented lattice through an embedding that respects the bounds of the original structure. The results obtained not only enrich the theoretical foundations of lattice theory but also provide potential avenues for applications in algebraic systems where unique complementation is an essential characteristic.