Joinirreducible

Joinirreducible is a concept in order theory and lattice theory describing a kind of indivisible element with respect to the join operation (least upper bound).

Definition and basic properties

An element j in a lattice L with a least element is joinirreducible if j is not the bottom element and whenever j equals a ∨ b for elements a and b of L, then j equals a or j equals b. In finite lattices, joinirreducible elements are exactly those that have a unique lower cover (they sit above precisely one element directly below them). Joinirreducibles play a key role in representing and decomposing lattice elements.

Examples

In the lattice of subsets of a set S, ordered by inclusion and with join given by union, the joinirreducible elements are the singleton subsets. Each nonempty finite subset is the union of its singleton components. In a Boolean algebra, the joinirreducibles are precisely the atoms (minimal nonzero elements).

Distributive lattices and representation

In any distributive lattice, every element can be expressed as a join of joinirreducibles below it; the joinirreducibles generate the lattice under joins. This leads to a foundational representation: Birkhoff’s representation theorem states that every finite distributive lattice is isomorphic to the lattice of order ideals (down-sets) of the poset of its joinirreducibles J(L). Under this correspondence, an element maps to the set of joinirreducibles beneath it, and joins correspond to unions.

Relation to related concepts

Join-prime is a stronger property than joinirreducible; every join-prime element is joinirreducible, but not conversely in general. The dual notion is meetirreducible, defined analogously with meets.

Applications

Joinirreducibles provide a convenient building-block framework for understanding the structure of lattices, especially finite distributive lattices, where they enable canonical decompositions and lattice reconstruction via posets of irreducibles.