On commutative unary algebras with the distributive congruations lattices

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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.

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Commutative unary algebra, �-unoid, latticies of congruences, distributive property, cyclic element

Короткий адрес: https://sciup.org/149129838

IDR: 149129838   |   DOI: 10.15688/mpcm.jvolsu.2018.3.2

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