Prove that if $(A,<)$ lattice and A is finite then A has maximun and minimun

Question

Let be A a finite set and (A,<) a lattice. Prove that A has maximun and minimun.

First, I'm from Uruguay and I'm literally translating the terms so this are the defitions I have, in order to avoid misconceptions:

Let be $R$ a partial order relation in $A$, and $B\subseteq A$:

Supremum: $x$ is supremum of $B$ if and only if $x$ is an upper bound of $B$ and $\forall y$ such that $y$ is upper bound of B, it holds $xRy$.

Maximun: If the supremum is included on $B$.

Lattice: R is a lattice if and only if $\forall B\subseteq A$, $B$ finite, it holds $B$ has supremum and infimum.

This is my reasoning:

Since A is a lattice, then it holds that $\forall B \subseteq A$ finite $B$ has supremum and infimum. Since $A$ is finite, we can choose $B = A$ and by lattice definition $A$ will have supremum and infimum. Then, that supremum must belong to $A$ (I intuitively think this but I don't know how to formalize it). The same with the infimum.

I always suffer with stuff related to proving anything so any help will be appreciated.

Answer 1

If $A$ is a finite lattice then by the definition of lattice given in your question $A$ has a supremum and has an infimum.

This means that elements $x,y\in A$ exist such that $y\leq a\leq x$ for every $a\in A$.

This means exactly that $x$ is a maximum and $y$ is a minimum of lattice $A$.

So your reasoning is correct.

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"Then, that supremum must belong to $A$ (I intuitively think this but I don't know how to formalize it)."

Of course it does. A supremum of a subset $B\subseteq A$ is by definition an element of $A$ that has a certain property.

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Footnote:

The definition of lattice given by you is often practicized as definition of bounded lattice which makes the question a bit more easy because every bounded lattice has a maximum and a minimum. For this take infimum and supremum of the empty set.

In a weaker definition it is only demanded that non-empty finite sets (or equivalently pairs of elements) have a supremum and an infinum. Also if that definition is practicized then it is possible to prove the existence of a minimum and a maximum if the lattice is finite. E.g. see the answer of William.

Answer 2

Let the points of A be $a_1, a_2,.. a_n$.
The maximum is $a_1 \vee a_2 \vee..\vee a_n$.
The minimum is $a_1 \wedge a_2 \wedge..\wedge a_n$.