A monoidal category need not be closed

In $M_3$, the meet operation satisfies $p\wedge q=0$, $p\wedge r=0$, and $p\wedge 1=p$, while $q$ and $r$ are incomparable below $1$. By [L1] and [L2], the associated poset-category is a strict monoidal category with tensor product $\wedge$ and unit $1$.

Fix $X:=p$ and $Y:=q$. An object $a$ satisfies $a\wedge p\le q$ exactly when $a\in \{0,q,r\}$: this holds for $0,q,r$, fails for $1$ because $1\wedge p=p\nleqq q$, and fails for $p$ because $p\wedge p=p\nleqq q$.

The set $\{0,q,r\}$ has no greatest element, since $q$ and $r$ are incomparable and both dominate $0$. Therefore there is no object $[p,q]$ with $a\le [p,q]$ iff $a\wedge p\le q$, so the functor $-\wedge p$ has no right adjoint. By [L3], this monoidal category is not right closed and hence not closed.