On partial $n$-ary groupoids whose equivalence relations are congruences

A. V. Reshetnikov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 232-239.

Résumé

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.

Détail

Keywords: partial $n$-ary groupoid, one-sided congruence, $R_i$-congruence, congruence lattice, equivalence relation lattice.

Bibliographie

[1] Kozhukhov I. B., Reshetnikov A. V., “Algebras whose equivalence relations are congruences”, Journal of Mathematical Sciences (New York), 177:6 (2011), 886–907 | DOI | MR | Zbl

[2] G. Grätzer, Universal algebra, 2nd ed. with updates, Springer, 2008 ; Second Edition, Springer Science+Business Media, LLC, 1979, 586 pp. | MR | Zbl | Zbl

[3] Ljapin E. S., Evseev A. E., The Theory of Partial Algebraic Operations, Springer Science + Business Media, B.V., 1997, 237 pp. | MR

[4] Grätzer G., Schmidt E. T., “Characterizations of congruence lattices of abstract algebras”, Acta Sci. Math. (Szeged), 24:3 (1963), 34–59 | MR

[5] G. Grätzer, G. H. Wenzel, “On the concept of congruence relation in partial algebras”, Math. Scand., 20 (1967), 275–280 | MR | Zbl

[6] Reshetnikov A. V., “On congruences of partial $n$-ary groupoids”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya: Matematika. Mekhanika. Informatika. (Izvestiya Saratovskogo universiteta. New series. Series: Mathematics. Mechanics. Informatics), 11:3/2 (2011), 46–51 (Russian)

[7] Berman J., “Strong congruence lattices of finite partial algebras”, Algebra Universalis, 1:1, December (1971), 133–135 | DOI | MR | Zbl

[8] Burmeister P., “Free partial algebras”, J. Reine Angew. Math., 1970:241, January (1970), 75–86 | MR | Zbl

[9] Clifford A. H., Hall T. E., “A characterisation of R-classes of semigroups as a partial groupoids”, Semigroup Forum, 6 (1973), 246–254 | DOI | MR | Zbl

[10] Kulik V. T., “O naibolshih silnyh otnosheniyah congruentnosti chastichnyh universalnyh algebr”, Issledovaniya po algebre, Izd. Saratovskogo Universiteta, Saratov, 1970, 40–46

[11] Kulik V. T., “O reshetkah silnyh otnosheniy congruentnosti polugruppoidov”, Uporyadochennye mnojestva i reshetki, 2, Izd. Saratovskogo Universiteta, Saratov, 1974, 42–50 | Zbl

[12] E. S. Ljapin, “Vnutrennee polugruppovoe prodoljenie nekotoryh polugruppovyh amalgam”, Izvestiya vuzov. Matematika, 1993, no. 11, 20–26 | MR

[13] Pastijn F., “A generalization of Green's equivalence relations for halfgroupoids”, Simon Stevin, 49 (1976), 165–175 | MR | Zbl

[14] Fleischer I., “On extending congruences from partial algebras”, Fund. math., 88 (1975), 11–16 | MR | Zbl

[15] Schelp R. H., “A partial semigroup approach to partially ordered sets”, Proc. London Math. Soc., 24 (1972), 46–58 | DOI | MR | Zbl