Сharacterization of distributive lattices of quasivarieties of unars

Let $L_q (\mathfrak{M})$ denote the lattice of all subquasivarieties of the quasivariety $\mathfrak{M}$ under inclusion. There is a strong correlation between the properties of the lattice $L_q (\mathfrak {M})$ and algebraic systems from $\mathfrak{M}$. A. I. Maltsev first drew attention to this fact in a report at the International Congress of Mathematicians in 1966 in Moscow.
In this paper, we obtain a characterization of the class of all distributive lattices, each of which is isomorphic to the lattice of some quasivariety of unars. A unar is an algebra with one unary operation. Obviously, any unar can be considered as an automaton with one input signal without output signals, or as an act over a cyclic semigroup.
We construct partially ordered sets $P_{\infty}$ and $P_s (s \in {\mathbf{N}_0})$, where ${\mathbf{N}_0}$ is the set of all non-negative integers. It is proved that a distributive lattice is isomorphic to the lattice $L_q (\mathfrak{M})$ for some quasivariety of unars $\mathfrak{M}$ if and only if it is isomorphic to some principal ideal of one of the lattices $O (P_s) (s \in {\mathbf{N}_0})$ or $O_c (P_{\infty})$, where $O (P_s) (s \in {\mathbf{N}_0})$ is the ideal lattice of the poset $P_s (s \in {\mathbf{N}_0})$ and $O_c (P_ {\infty})$ is the ideal lattice with a distinguished element $c$ of the poset $P _ {\infty}$.
The proof of the main theorem is based on the description of $\mathrm{Q}$-critical unars. A finitely generated algebra is called $\mathrm{Q}$-critical if it does not decompose into a subdirect product of its proper subalgebras. It was previously shown that each quasivariety of unars is determined by its $\mathrm{Q}$-critical unars. This fact is often used to investigate quasivarieties of unars.