New theorem and conjectures

I have a little generalized the following old theorem:

$(a\sqcap^{\mathfrak{A}}b)^{\ast}=(a\sqcap^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcup^{\mathfrak{A}}b^{\ast}=a^{+}\sqcup^{\mathfrak{A}}b^{+}$ .

I have also found a new (easy to prove) theorem:

$(a\sqcup^{\mathfrak{A}}b)^{\ast}=(a\sqcup^{\mathfrak{A}}b)^{+}=a^{\ast}\sqcap^{\mathfrak{A}}b^{\ast}=a^{+}\sqcap^{\mathfrak{A}}b^{+}$ .

The above formulas hold for filters on a set (and some generalizations).

Do these formulas hold also for funcoids? (an interesting conjecture)