Join-irreducible element

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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.


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