On maximal clones of ultrafunctions of rank 2

This paper considers functions mapping a 2-element set $A$ to all non-empty subsets of $A$. These functions are called ultrafunctions of rank 2. Ultrafunctions of rank 2 can be interpreted as functions on all non-empty subsets of $A$. Value of ultrafunction on set $B \subseteq A$ is determined as intersection of values on all elements of $B$, if this intersection is not empty, and as union of these values otherwise. Thus an unltrafunction can be specified by all of its values on elements of $A$. Superposition of ultrafunctions is determined the same way.
The number of maximal clones for all ultrafunctions of rank 2 is equal to 11 [V. Panteleev, 2009].
This paper studies properties of ultrafunctions with respect of their inclusion in maximal clones $\mathbb {K}_5$, $\mathbb {S}^-$, $\mathbb {T}_0^-$ and $\mathbb {T}_1^-$. These properties give some results concerning clone lattice (e.g., clones of intervals $I(\mathbb {T}_0\cap \mathbb {T}_1,\mathbb {T}_0)$ and $I(\mathbb {T}_0\cap \mathbb {T}_1,\mathbb {T}_1)$ are not included in clone $\mathbb {S}^-$; all self-dual and monotone ultrafuncions are included in $\mathbb {K}_1$ and $\mathbb {K}_2$). Some borders on classes of equivalence number are described (ultrafunctions not included in clones $\mathbb {T}_1^-$ and $\mathbb {K}_5$ generate no more than 32 classes of equivalence by relation of belonging to maximal clones). These results can be applied to classification of ultrafunctions by their inclusion in maximal clones.