Converse to a proposition on lattices and join-irreducible elements

lattice-orders

Let $(L;\leq)$ be a lattice. I know that if $L$ is linearly ordered, then every element except the bottom one if there is any is join irreducible. Is the converse of that proposition true? That is, given that $L$ is a lattice, if we know that every non-bottom element of $L$ is join irreducible, must $L$ be linearly ordered?


Solution 1:

Yes. Suppose $x,y$ are incomparable in a lattice. Then both $x,y<x\lor y$, so the element $x\lor y$ is not join-irreducible.

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