Join-irreducible element

in a lattice $L$

An element $a$ which is not the minimum element $0_L$ (if any) and for which $x < a$ and $y < a$ implies $x \vee y < a$. The latter condition is equivalent to $a = x \vee y$ implies $a = x$ or $a = y$.

Dually, a meet-irreducible element $t$ is not the maximum element $1_L$ if any and $t < x,\,t<y \Rightarrow t < x \wedge y$ or $t = x \wedge y \Rightarrow t = x \,\text{or}\, t = y$. An element which is both join-irreducible and meet-irredicuble is doubly-irreducible.

In a lattice satisfying the descending chain condition, in particular for a finite lattice, the join-irreducible elements are those which cover precisely one element; further, every element is a join of join-irreducible elements.

In the lattice of natural numbers ordered by divisibility, the join-irreducible elements are the prime powers.

References

• B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001
• Rota, Gian-Carlo (with P. Doubilet, C. Greene, D. Kahaner, A: Odlyzko and R. Stanley) Finite operator calculus Academic Press (1975) ISBN 0-12-596650-4 Zbl 0328.05007