On commutative unary algebras with the distributive congruations lattices

The article is devoted to the study of lattices of congruences of unary algebras. Algebras with unary operations were considered by A. I. Mal-tsev [6, p. 348] and were called �-unoids. Unar is an algebra with one unaryoperation.In [2; 3; 11] unars whose congruence lattices belong to a given class of lattices (semimodular, atomic, distributive, etc.) were studied. Similar questions for unary algebras with two unary operations were considered in [7; 8; 10]. Important results on commutative unary algebras with a distributive lattice congruences were obtained in [4; 5]. The main results of this note is announced in [9].∈∈ ∈The unary algebra A = (𝐴, Ω) is an algebraic system, which is defined by some set and a set Ω of unary operations on 𝐴. Each operation Ω can be considered as a mapping of the set into itself. The algebra A = (𝐴, Ω) is said to be commutative if for all 𝑓, Ω and all 𝐴 it holds the equality𝑓 (𝑔(�)) = 𝑔(𝑓 (�)).∈ ∈∈ ∈≤The congruence θ on the algebra A is such an equivalence relation on 𝐴, that for each Ω and all �, 𝐴 from �θ� it follows (�)θ𝑓 (�). By Con A is denoted the set of all congruences on algebra A. There is a partial order on Con A: for the congruences θ1, θ2 the relation θ1 θ2 is satisfied if and only∧if for any elements �, 𝐴 from �θ1� it follows �θ2�. If θ1, θ2 Con A,then θ1 θ2 denotes the lower bound congruences θ1 and θ2, then is the largest∈ ≤ ≤ ∨congruence θ Con A for which θ θ1 and θ θ2. The upper bound θ1 θ2of congruences θ1 and θ2.∈ ∧ ∨ ∧ ∨ ∧A lattice of congruences Con A is called distributive if for any three congruences θ1, θ2, θ3 Con A the equality θ1 (θ2 θ3) = (θ1 θ2) (θ1 θ3).Below we need the description of the following unars and unary algebras:Example 1 The unar D1 is (N, ), where N is the set of natural numbers, and the operation is defined by the formula (�) = + 1, ∈ N.∈≥Example 2 For natural numbers 1, the unar D2 is (Z�, ), where Z� is the residue ring modulo and (�) = + 1(mod �) for Z�. If, in addition, = 1, then the unary carrier consists of a single element, and is the identity map.Example 3 The unary algebra D3 is (Z, 𝑓, 𝑔), where Z is the set of integers, and 𝑓, are defined by formulas (�) = + 1 and 𝑔(�) = - 1,� ∈ Z.The main result of this note is as follows: | | ≥Theorem 1. Let A = (𝐴, Ω) be a commutative unary algebra with a distributive lattice of congruences, = Ω 2. Then this algebra containsa subalgebra, the lattice of congruences of which is isomorphic to the lattice of congruences one of the unars D1, D2(�) or a lattice congruences of the algebra D3.