On partial $n$-ary groupoids whose equivalence relations are congruences
A. V. Reshetnikov National Research University of Electronic Technology
Abstract:
G. Grätzer's gives the following example
in his monograph «Universal algebra».
Let
$A$ be a universal algebra
(with some family of operations
$\Sigma$).
Let us take an arbitrary set
$B \subseteq A$.
For all of the operations
$f \in \Sigma$
(let
$n$ be the arity of
$f$)
let us look how
$f$ transformas the elements of
$B^{n}$.
It is not necessary that
$f(B) \subseteq B$,
so in the general case
$B$ is not a subalgebra of
$A$.
But if we define partial operation as mapping from
a subset of the set
$B^n$ into the set
$B$.
then
$B$ be a set with a family of partial operations defined on it.
Such sets are called partial universal algebras.
In our example
$B$ will be a partial universal subalgebra of
the algebra
$A$, which means the set
$B$ will be closed
under all of the partial operations of the partial algebra
$B$.
So, partial algebras can naturally appear when studying
common universal algebras.
The concept of congruence of universal algebra can be generalized
to the case of partial algebras.
It is well-known that the congruences of a partial universal
algebra
$A$ always from a lattice, and if
$A$ be a full algebra
(i.e. an algebra) then the lattice of the congruences of
$A$ is
a sublattice of the lattice of the equivalence relations on
$A$.
The congruence lattice of a partial universal algebra is its
important characteristics.
For the most important cases of universal algebra
some results were obtained which characterize the algebras
$A$
without any congruences except the trivial congruences
(the equality relation on
$A$ and the relation
$A^2$).
It turned out that in the most cases, when the congruence
lattice of a universal algebra is trivial the algebra itself
is definitely not trivial.
And what can we say about the algebras
$A$ whose equivalence relation
is, vice versa, contains all of the equivalence relations on
$A$?
It turns out, in this case any operation
$f$ of the algebra
$A$
is either a constant (
$|f(A)| = 1$) or a projection
(
$f(x_1,$ …,
$x_i$, …,
$x_n) \equiv x_i$).
Kozhukhov I. B. described the semigroups whose equivalence relations
are one-sided congruences. It is interesting now to generalize
these results to the case of partial algebras.
In this paper the partial
$n$-ary groupoids
$G$ are studied
whose operations
$f$ satisfy the following condition:
for any elements
$x_1$, …,
$x_{k-1}$,
$x_{k+1}$, …,
$x_n \in G$
the value of the expression
$f(x_1$, …,
$x_{k-1}$,
$y$,
$x_{k+1}$, …,
$x_n)$
is defined for not less that three different elements
$y \in G$.
It will be proved that if any of the congruence relations on
$G$
is a congruence of the partial
$n$-ary groupoid
$(G,f)$
then under specific conditions for
$G$ the partial operation
$f$
is not a constant.
Bibliography: 15 titles.
Keywords:
partial $n$-ary groupoid, one-sided congruence, $R_i$-congruence, congruence lattice, equivalence relation lattice.
UDC:
512.548.2 +
512.571 Received: 21.12.2015
Accepted: 11.03.2016