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I have an explanation here as to why the GLB of two elements in a poset is unique whenever it exists... but I can't quite understand the concept of GLB and LUB.

Suppose $x$ and $y$ are each glbs of two elements $a$ and $b$. Then $x\preceq a$, $x\preceq b$ implies $x\preceq y$ because $y$ is a greatest lower bound, and $y\preceq a$, $y\preceq b$ implies $y\preceq x$ is greatest. So by antisymmetry, $x=y$.

Looking at this, I have to ask why $x\preceq a\wedge x\preceq b\Rightarrow x\preceq y$.

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1 Answer 1

up vote 2 down vote accepted

Since $x\preceq a$ and $x\preceq b$, then $x$ is a lower bound of $a,b$, but $y$ is a greatest lower bound of $a,b$, so must be "at least as big as" $x$, meaning $x\preceq y$. That's just part of the definition. Namely, $z$ is a greatest lower bound of $a,b$ if and only if:

(i) $z\preceq a$ and $z\preceq b$ ($z$ is a lower bound of $a,b$), and

(ii) for any lower bound $z'$ of $a,b$, we have $z'\preceq z$ (this is the "greatest" part).

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