How to prove that an ordered set is a lattice.

Question

I am given an interval $a,b$ with $a \le x$ and $x \le b$ . How can I prove that this let s call it $( L ,\le)$ is a lattice ? I know it is probably stupidly simple but I would appreciate some help .

Answer 1

According to @amrsa, your question is incomplete. I've answered this by assuming $(L,\le)$ is a chain.

Use the definition. Let $A=\{x_1,x_2\}\subseteq L$. Then you need to show two things.

1. $A$ has an infimum in $L$
2. $A$ has a supremum in $L$

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For infimum, you need to show that $\exists i \in L$ such that

• $i \le x_1$ and $i \le x_2$
• $\forall x\in L$ if $x \le x_1$ and $x \le x_2$, then $x\le i$

For supremum, show that $\exists s \in L$ such that

• $x_1\le s$ and $x_2\le s$
• $\forall x\in L$ if $x_1\le x$ and $x_2\le x$, then $s\le x$

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Hint:

Choose $i=\min(x_1,x_2)$ and $s=\max(x_1,x_2)$ and show that they satisfied the above criteria.