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

  • Thread starter Thread starter QuestForInsight
  • Start date Start date
  • Tags Tags
    Bound Upper bound
QuestForInsight
Messages
34
Reaction score
0
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?
 
Physics news on Phys.org
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.
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Replies
3
Views
5K
Replies
4
Views
2K
Replies
2
Views
8K
Replies
9
Views
2K
Replies
14
Views
3K
Replies
56
Views
3K
Back
Top