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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 232–239 (Mi cheb466)

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.

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UDC: 512.548.2 + 512.571
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 232--239 \mathnet{http://mi.mathnet.ru/cheb466} \elib{https://elibrary.ru/item.asp?id=25795086}