Сharacterization of distributive lattices of quasivarieties of unars

Abstract

Let L_q(M) denote the lattice of all subquasivarieties of the quasivariety M under inclusion. There is a strong correlation between the properties of the lattice L_q(M) and algebraic systems from 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∞ and P_s(s ∈ N0), where N0 is the set of all nonnegative
integers. It is proved that a distributive lattice is isomorphic to the lattice L_q(M) for some quasivariety of unars M if and only if it is isomorphic to some principal ideal of one of the lattices 0(Ps)(s ∈ N0) or 0c(P∞), where 0(Ps)(s ∈ N0) is the ideal lattice of the poset Ps(s ∈ N0) and 0s(P∞) is the ideal lattice with a distinguished element 𝑐 of the poset P∞.
The proof of the main theorem is based on the description of Q-critical unars. A finitely generated algebra is called 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 Q-critical unars. This fact is often used to investigate quasivarieties of unars.