# Lattices: What does it mean that every pair of elements has an infimum and a supremum?

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.

• for positive integers, every pair has a $\gcd$ and an $\operatorname{lcm}$ Commented Jan 23, 2021 at 19:34
• thanks for your answer but excuse my ignorance or language barrier, but what is ged and lem ? best wishes Commented Jan 23, 2021 at 19:42
• I'm not certain it's a language barrier.
– user436658
Commented Jan 23, 2021 at 20:03
• @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. Commented Jan 24, 2021 at 9:58

## 1 Answer

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).

• Thank you ! ("2 more to go") Commented May 19, 2021 at 9:29