# Join and meet

Let $$\langle P, \leq \rangle$$ be a par­tially or­dered set, and let $$S \subseteq P$$. The join of $$S$$ in $$P$$, de­noted by $$\bigvee_P S$$, is an el­e­ment $$p \in P$$ satis­fy­ing the fol­low­ing two prop­er­ties:

• p is an up­per bound of $$S$$; that is, for all $$s \in S$$, $$s \leq p$$.

• For all up­per bounds $$q$$ of $$S$$ in $$P$$, $$p \leq q$$.

$$\bigvee_P S$$ does not nec­es­sar­ily ex­ist, but if it does then it is unique. The no­ta­tion $$\bigvee S$$ is typ­i­cally used in­stead of $$\bigvee_P S$$ when $$P$$ is clear from con­text. Joins are of­ten called least up­per bounds or supre­mums. For $$a, b$$ in $$P$$, the join of $$\{a,b\}$$ in $$P$$ is de­noted by $$a \vee_P b$$, or $$a \vee b$$ when $$P$$ is clear from con­text.

The dual con­cept of the join is that of the meet. The meet of $$S$$ in $$P$$, de­noted by $$\bigwedge_P S$$, is defined an el­e­ment $$p \in P$$ satis­fy­ing.

• p is a lower bound of $$S$$; that is, for all $$s$$ in $$S$$, $$p \leq s$$.

• For all lower bounds $$q$$ of $$S$$ in $$P$$, $$q \leq p$$.

Meets are also called in­fi­mums, or great­est lower bounds. The no­ta­tions $$\bigwedge S$$, $$p \wedge_P q$$, and $$p \wedge q$$ are all have mean­ings that are com­pletely analo­gous to the afore­men­tioned no­ta­tions for joins.

## Ba­sic example

The above Hasse di­a­gram rep­re­sents a poset with el­e­ments $$a$$, $$b$$, $$c$$, and $$d$$. $$\bigvee \{a,b\}$$ does not ex­ist be­cause the set $$\{a,b\}$$ has no up­per bounds.$$\bigvee \{c,d\}$$ does not ex­ist for a differ­ent rea­son: al­though $$\{c, d\}$$ has up­per bounds $$a$$ and $$b$$, these up­per bounds are in­com­pa­rable, and so $$\{c, d\}$$ has no least up­per bound. There do ex­ist sub­sets of this poset which pos­sess joins; for ex­am­ple, $$a \vee c = a$$, $$\bigvee \{b,c,d\} = b$$, and $$\bigvee \{c\} = c$$.

Now for some ex­am­ples of meets.$$\bigwedge \{a, b, c, d\}$$ does not ex­ist be­cause $$c$$ and $$d$$ have no com­mon lower bounds. How­ever, $$\bigwedge \{a,b,d\} = d$$ and $$a \wedge c = c$$.