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24-Jan-2019 22:16

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I’ll use the following Hasse diagram of a partial order (taken from this question) as an example.It isn’t the Hasse diagram of a lattice, but it’s fine for illustrating greatest lower bounds and least upper bounds. What elements $x$ have the property that $n\le x$ and $g\le x$? It’s not hard to see that $r$ is one of them: you can get from $n$ to $r$ by travelling upwards in the diagram, and you can also get from $g$ to $r$ by travelling upwards in the diagram.If $x$ and $y$ are elements of a partial order, an upper bound for $x$ and $y$ is simply an element $u$ such that $x\le u$ and $y\le u$; $u$ is the least upper bound of $x$ and $y$ if $u$ is $\le$ all upper bounds of $x$ and $y$. Similarly, $s,t$, and $u$ are upper bounds for $n$ and $g$.There are two more that are a little harder to spot: $d$ and $a$ are also upper bounds for $n$ and $g$, for the same reason: you can get to each of them from both $n$ and $g$ by travelling upwards in the diagram.On the other hand, $i$ is not an upper bound for $n$ and $g$: $n\le i$, but $g\not\le i$.Now look at the set of upper bounds for $n$ and $g$: it’s $$.

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