lintervals

Lintervals are a concept in abstract algebra, specifically within the study of lattices. A lattice is a partially ordered set where every pair of elements has a unique least upper bound (join) and a unique greatest lower bound (meet). Lintervals, often denoted as [a, b], represent a specific type of subset within a lattice. The interval [a, b] is defined as the set of all elements x in the lattice such that a is less than or equal to x and x is less than or equal to b. This means that for any element x within the interval [a, b], it must be both greater than or equal to the lower bound 'a' and less than or equal to the upper bound 'b' according to the lattice's partial order.

The concept of linters is fundamental in understanding the structure of lattices. The set of all linters within a lattice forms a new lattice itself, known as the lattice of intervals. This lattice of intervals has its own join and meet operations, which are defined based on the operations in the original lattice. For example, the join of two intervals [a, b] and [c, d] is the smallest interval that contains both. The meet of two intervals is similarly defined as the largest interval contained within both. The study of linters allows mathematicians to explore substructures of lattices and analyze their properties in a more granular way.