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

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In a partially ordered set P, an element c is defined as the greatest upper bound of elements a and b if for every x in L, x ≤ c if and only if x ≤ a and x ≤ b. Conversely, c is the least upper bound if c ≤ x for every x in L if and only if a ≤ x and b ≤ x. The confusion arises from the sudden introduction of L, which some participants believe should refer back to P. Additionally, there was a mix-up regarding terminology, with "greatest upper bound" mistakenly referred to as "greatest lower bound." The definitions ultimately relate to the concept of a lattice, where L is properly defined later in the text.
QuestForInsight
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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?
 
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QuestForInsight said:
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.
 
Opalg said:
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.
 

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