The article is devoted to the study of
lattices of congruences of unary algebras.
Algebras with $m$ unary operations were considered by
A. I. Maltsev [6, p. 348]
and were called $m$-unoids. Unar is an algebra with one unary
operation.
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$ = \left (A, \Omega \right)$
is an algebraic system,
which is defined by some set $A$ and a set $\Omega$
of unary operations on $A$.
Each operation $f\in\Omega$ can be considered as a mapping of the set
$A$ into itself.
The algebra A$ = \left (A,\Omega\right)$
is said to be commutative if for all $f, g \in \Omega$
and all $x\in A$ it holds the equality
$f(g(x)) = g(f(x))$.
The congruence $\theta$ on the algebra A is such an
equivalence relation on $A$, that
for each $f\in\Omega$ and all $x,y \in A$ from $x\theta y$
it follows $f(x)\theta f(y)$.
By ${\rm Con} \ $Ais denoted the set of all congruences
on algebra A.
There is a partial order on ${\rm Con} \ $A:
for the congruences $\theta_1, \theta_2$ the relation
$\theta _1 \leq \theta _2$ is satisfied if and only if
for any elements $x, y \in A$ from $x \theta_1 y$
it follows $x \theta_2 y $.
If $\theta_1, \theta_2 \in {\rm Con} \ $A, then
$\theta_1 \wedge \theta_2 $ denotes the lower bound
congruences $\theta_1 $ and $\theta_2 $, then
is the largest congruence $\theta \in {\rm Con} \ $A for which
$\theta \leq \theta_1 $ and $\theta \leq \theta_2 $.
The upper bound $\theta_1 \vee \theta_2 $ of congruences
$\theta_1 $ and $\theta_2 $.
A lattice of congruences ${\rm Con} \ $A is called distributive if
for any three congruences
$\theta_1, \theta_2, \theta_3 \in {\rm Con} \ $A the equality
$\theta_1 \wedge (\theta_2 \vee \theta_3) =
(\theta_1 \wedge \theta_2) \vee (\theta_1 \wedge \theta_3) $.
Below we need the description of the following unars and unary algebras:
Example 1
The unar D$_1$ is (N, $f$), where
N is the set of natural numbers, and the operation $f $ is defined
by the formula $f (x) = x + 1 $, $x \in $N .
Example 2 For natural numbers $n\geq 1$,
the unar D$_2$ is (Z$_n, f $), where
Z$_n$ is the residue ring modulo $n$
and $f (x) = x + 1 ($mod$\ n) $
for $x \in $Z$_n $.
If, in addition, $n = 1$, then the unary carrier consists of a single element,
and $f$ is the identity map. Example 3 The unary algebra D$_3$ is (Z,$ f,g$), where
Z is the set of integers, and $f$, $g$
are defined by formulas
$f (x) = x + 1 $ and $g (x) = x-1 $, $x \in \mathbf{Z} $.
The main result of this note is as follows:
Theorem 1.
Let A $= \left (A, \Omega \right) $ be a commutative unary
algebra with a distributive lattice of congruences, $m = |\Omega| \geq 2 $.
Then this algebra contains a subalgebra,
the lattice of congruences of which is isomorphic to the lattice of congruences
one of the unars D$_1 $, D$_2 (n) $ or a lattice
congruences of the algebra D$_3 $.
