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what does it mean that every relation of two (or more?*) elements has an infimum or maximum?

Could you please give me some examples?

Thanks von Spotz

* while, so far as i know, the operations of both the order-relation interpretation as well as the algebraic-structure interpretation of a lattice are binary only.

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    $\begingroup$ for positive integers, every pair has a $\gcd$ and an $\operatorname{lcm}$ $\endgroup$
    – Will Jagy
    Commented Jan 23, 2021 at 19:34
  • $\begingroup$ thanks for your answer but excuse my ignorance or language barrier, but what is ged and lem ? best wishes $\endgroup$
    – von spotz
    Commented Jan 23, 2021 at 19:42
  • $\begingroup$ I'm not certain it's a language barrier. $\endgroup$
    – user436658
    Commented Jan 23, 2021 at 20:03
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    $\begingroup$ @vonspotz $\gcd$ stands for greatest common divisor and $\mathrm{lcm}$ for least common multiple. These, considered as operations on the natural numbers, endow this set with the structure of lattice, in which the associated order is the one of divisibility. $\endgroup$
    – amrsa
    Commented Jan 24, 2021 at 9:58

1 Answer 1

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Let $P$ be a poset ordered by $\leq$. For $a, b \in P$ we say that $c$ is a supremum of $a$ and $b$ if $a \leq c$ and $b \leq c$ and furthermore for every $d \in P$ such that $a \leq d$ and $b \leq d$ we have $c \leq d$. In words this means that a supremum of two elements is greater than both elements, so it is an upper bound, and it the least upper bound.

The notion of infimum is dual: $c$ is the infimum of $a$ and $b$ if $c \leq a$ and $c \leq b$ while for every $d \in P$ with $d \leq a$ and $d \leq b$ we have $d \leq c$.

Exercise. The supremum of $a$ and $b$, if it exists, is unique. That is, if $c$ and $c'$ are both the supremum of $a$ and $b$ then $c = c'$. Likewise the infimum is unique.

We call a poset $P$ a lattice if any two elements have both a supremum and infimum. For the supremum of $a$ and $b$ we write $a \vee b$ (by the exercise this makes sense because there is a unique element with this role). For the infimum of $a$ and $b$ we write $a \wedge b$.

We can generalise things to finite subsets. For example: for a finite subsets $\{a_1, \ldots, a_n\} \subseteq P$ we say that $c$ is a supremum of $\{a_1, \ldots, a_n\}$ if $a_i \leq c$ for every $1 \leq i \leq n$ and for any $d$ with $a_i \leq c$ for all $1 \leq i \leq n$ we have $c \leq d$. Similarly for infimum.

Exercise. A poset $P$ is a lattice if and only if every nonempty finite subset has a supremum and an infimum. The right to left direction should be trivial. For the left to right direction you can use the hint that given $\{a_1, \ldots, a_n\}$ we can consider $(((a_1 \vee a_2) \vee a_3) \vee \ldots) \vee a_n$ (to be precise: use induction).

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  • $\begingroup$ Thank you ! ("2 more to go") $\endgroup$
    – von spotz
    Commented May 19, 2021 at 9:29

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