Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $x\preceq a\wedge x\preceq b\Rightarrow x\preceq y$.

share|improve this question
add comment

1 Answer

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.