partial order and truth value

Question

please,give a formula Q with free variables such that for any partial order ( A ; ≤ ; -)

we have that [Q]=1 iff [z] is a join of [x] and [y]

Answer 1

HINT: I’m assuming that $[x]$ is the interpretation of $x$ in the partial order, while $[Q]$ is the truth value of $Q$. You want a formula $Q(x,y,z)$ that says that $z$ is the join of $x$ and $y$, but you have only the order $\le$ to work with: you don’t have the join operation. If $a,b,c\in A$, $c$ is the join of $a$ and $b$ precisely when $c$ is the least upper bound of the set $\{a,b\}$. This means two things:

• $c$ is an upper bound for $\{a,b\}$; and
• $c$ is the least upper bound for $\{a,b\}$: if $u$ is any upper bound for $\{a,b\}$, then $c\le u$.

Your formula $Q(x,y,z)$ therefore has to say that $z$ is an upper bound for $x$ and $y$ and that it’s the least of the upper bounds. Just translate these requirements into logical formalism, and you’ll have $Q(x,y,z)$.