Does transitive closure necessarily exists? (in an abstract setting)

Question

Consider a set $F$ which is both monoid (with identity $1$ and composition $\circ$) and complete lattice with join $\bigsqcup$ and order $\leq$.

Let the following identities hold:

1. $(a\circ b)\sqcup c=(a\circ c)\sqcup(b\circ c)$;
2. $c\sqcup(a\circ b)=(c\circ a)\sqcup(c\circ b)$.

Let $\mu\in F$.

I name the minimal reflexive ($x\geq 1$) transitive ($x\circ x\leq x$) element $x$ which is above $\mu$ as transitive closure of $\mu$.

Are the following statements necessarily true?

• Transitive closure of $\mu$ exists.

• Transitive closure is the recursively defined set with elements defined as:

  1. $1$;
  2. $\mu$;
  3. composition of other elements of the set;
  4. finite and infinite (countable) joins of other elements of the set.