View source for 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}}

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