Lattice orders and number of elements in a set

Question

My discrete mathematics lecture notes give the following definition of a lattice order:

A 'Partial order R is a lattice order if the set of lower bounds for any two elements $x, y ∈ X$ has the maximum element and the set of upper bounds for any two elements $x, y ∈ X$ has the minimum element.'

I assumed that $x, y$ must be distinct, especially after having looked at the following definition of a lattice-ordered set and noted that a set is a collection of distinct objects. However, my lecturer holds to his claim that the elements $x, y$ above do not have to be distinct.

Is this possible - i.e., is it possible for $x, y$ in the definition above to be the same?

Answer 1

Really the thing here to notice is that $x\wedge x=x$ and $x\vee x=x$:

Certainly $x$ is less than or equal to $x$ and so is a lower bound of $\{x\}$. Moreover, if $x<y<x$, we reach a contradiction. Thus, $x$ is the \emph{greatest} lower bound of $\{x\}$. Thus, $x\wedge x=\inf\{x,x\}=x$

Analogous reasoning shows that $x$ is the least upper bound of $\{x\}$. Thus, $x\vee x=\sup\{x,x\}=x$.

Answer 2

If $x=y$ the set of lower bounds is $L_x=\{z\in X: z\le x\}$. Since $x\in L_x$ and $x$ is an upper bound for $L_x$, $x=\max L_x$.

We see that the set of lower bounds of one element has always a maximum; analogously, the set of upper bounds of one element has a minimum. In both cases the max and the min is the element itself.

Imposing or not the condition $x\neq y$ in your definition is irrelevant.