# Join and meet: Exercises

Try these ex­er­cises to test your knowl­edge of joins and meets.

## Tan­gled up

Deter­mine whether or not the fol­low­ing joins and meets ex­ist in the poset de­picted by the above Hasse di­a­gram.

$$c \vee b$$

This join does not ex­ist, be­cause $$i$$ and $$j$$ are in­com­pa­rable up­per bounds of $$\{ c, b \}$$, and no smaller up­per bounds of $$\{ c, b \}$$ ex­ist.

$$g \vee e$$

This join ex­ists. $$g \vee e = j$$.

$$j \wedge f$$

This meet does not ex­ist, be­cause $$d$$ and $$b$$ are in­com­pa­rable lower bounds of $$\{ j, f \}$$, and no larger lower bounds of $$\{ j, f \}$$ ex­ist.

$$\bigvee \{a,b,c,d,e,f,g,h,i,j,k,l\}$$

This join ex­ists. It is $$l$$.

## Join fu

Let $$P$$ be a poset, $$S \subseteq P$$, and $$p \in P$$. Prove that if both $$\bigvee S$$ and $$(\bigvee S) \vee p$$ ex­ist then $$\bigvee (S \cup \{p\})$$ ex­ists as well, and $$(\bigvee S) \vee p = \bigvee (S \cup \{p\})$$.

For any $$X \subset P$$, let $$X^U$$ de­note the set of up­per bounds of $$X$$. The above propo­si­tion fol­lows from the fact that $$\{\bigvee S, p\}^U = (S \cup p)^U$$, which is ap­par­ent from the fol­low­ing chain of bi-im­pli­ca­tions:

$$q \in \{\bigvee S, p\}^U \iff$$

for all $$s \in S, q \geq \bigvee S \geq s$$, and $$q \geq p \iff$$

$$q \in (S \cup \{p\})^U$$.

If two sub­sets of a poset have the same set of up­per bounds, then ei­ther they both lack a least up­per bound, or both have the same least up­per bound. <div><div>

## Meet fu

Let $$P$$ be a poset, $$S \subseteq P$$, and $$p \in P$$. Prove that if both $$\bigwedge S$$ and $$(\bigwedge S) \wedge p$$ ex­ist then $$\bigwedge (S \cup \{p\})$$ ex­ists as well, and $$(\bigwedge S) \wedge p = \bigwedge(S \cup \{p\})$$.

Note that the propo­si­tion we are try­ing to prove here is the dual of the one stated in join fu. Thanks to the du­al­ity prin­ci­ple, this the­o­rem there­fore comes for free with our solu­tion to join fu.

## Quite a big join

In the poset $$\langle \mathbb N, | \rangle$$ dis­cussed in Poset: Ex­am­ples, does $$\bigvee \mathbb N$$ ex­ist? If so, what is it?