What is the definition of greatest/least upper bound in a partially ordered set?

[QuestForInsight]

Let $a$, $b$ and $c$ be elements of a partially ordered set $P$. My book defines $c$ as the greatest upper bound of $a$ and $b$ if, for each $x \in L$, we have $x \le c$ if and only if $x \le a$ and $x \le b$. Similarly, it defines $c$ as the least upper bound of $a$ and $b$ if, for each $x \in L$, we have $ c \le x$ if and only if $ a \le x$ and $b \le x$.

The thing is, the L appeared out of nowhere and the definition only makes sense to me if L was P. What do you think?

 

[Opalg]

Let $a$, $b$ and $c$ be elements of a partially ordered set $P$. My book defines $c$ as the greatest upper bound of $a$ and $b$ if, for each $x \in L$, we have $x \le c$ if and only if $x \le a$ and $x \le b$. Similarly, it defines $c$ as the least upper bound of $a$ and $b$ if, for each $x \in L$, we have $ c \le x$ if and only if $ a \le x$ and $b \le x$.

The thing is, the L appeared out of nowhere and the definition only makes sense to me if L was P. What do you think?

I agree, it seems that the author has switched from P to L without realising it. Another error is that "greatest upper bound" should be "greatest lower bound". Other than that, the definitions are correct.

 

[QuestForInsight]

I agree, it seems that the author has switched from P to L without realising it. Another error is that "greatest upper bound" should be "greatest lower bound". Other than that, the definitions are correct.

Many thanks. That other error was mine, sorry. This definition was part of the definition of lattice and few paragraphs later he denotes a lattice by L. So that probably explains the slip.