Let $(X,\le)$ be a bounded lattice and $a,b\in X$ and $a\vee b$ and $a\wedge b$ have complements. Do $a$ and $b$ have complements necessarily?
2026-04-01 16:49:51.1775062191
$a\vee b$ and $a\wedge b$ have complements. Does $a$ have one?
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Not necessarily. Consider the $7$-element lattice consisting of the numbers $1,2,3,5,6,10,30$ ordered by divisibility. Now $2\vee3=6$ has a unique complement, namely $5$; and $2\wedge3=1$ has a unique complement, namely $30$; and $3$ has two complements, $5$ and $10$; but $2$ has no complement at all.