Meetirreducibles

Meetirreducibles are a concept in abstract algebra, specifically within the study of lattices. In a distributive lattice, an element is called meetirreducible if it is not the bottom element and it can only be expressed as the meet of two other elements if at least one of those elements is equal to the meetirreducible element itself. More formally, an element $a$ in a lattice $L$ is meetirreducible if $a \neq \mathbf{0}$ (where $\mathbf{0}$ is the bottom element of $L$) and whenever $a = x \wedge y$ for some $x, y \in L$, it must be that $a = x$ or $a = y$.

This definition is closely related to the concept of joinirreducibles, which are defined dually. In a distributive lattice, the set of all elements can be uniquely decomposed into a meet of joinirreducibles and a join of meetirreducibles. This property is fundamental in understanding the structure of distributive lattices and is a key element in the representation theorem for distributive lattices, which states that every distributive lattice is isomorphic to a sublattice of a power set algebra.

The concept of meetirreducibles is particularly important in the study of Boolean algebras, which are a special type of distributive lattice. In a Boolean algebra, meetirreducible elements correspond to atoms, which are the minimal non-zero elements. The structure of a finite Boolean algebra is directly determined by its atoms. The notion of meetirreducibility also finds applications in logic and theoretical computer science, particularly in areas dealing with formal concept analysis and rule learning.