
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 wellknown 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 onesided 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
$x_1$, …, $x_{k1}$, $x_{k+1}$, …, $x_n \in G$
the value of the expression
$f(x_1$, …, $x_{k1}$, $y$, $x_{k+1}$, …, $x_n)$
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, onesided congruence, $R_i$congruence, congruence lattice, equivalence relation lattice.
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UDC:
512.548.2 + 512.571 Received: 21.12.2015 Accepted:11.03.2016
Citation:
A. V. Reshetnikov, “On partial $n$ary groupoids whose equivalence relations are congruences”, Chebyshevskii Sb., 17:1 (2016), 232–239
Citation in format AMSBIB
\Bibitem{Res16}
\by A.~V.~Reshetnikov
\paper On partial $n$ary groupoids whose equivalence relations are congruences
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 232239
\mathnet{http://mi.mathnet.ru/cheb466}
\elib{https://elibrary.ru/item.asp?id=25795086}
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http://mi.mathnet.ru/eng/cheb466 http://mi.mathnet.ru/eng/cheb/v17/i1/p232
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