# 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

**How to Cite This Entry:**

Join-irreducible element.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Join-irreducible_element&oldid=38971