ON PARTIAL n-ARY GROUPOIDS WHOSE EQUIVALENCE RELATIONS ARE CONGRUENCES

G. Gr¨atzer’s gives the following example in his monograph «Universal algebra». Let A be a universal algebra (with some family of operations Σ). Let us take an arbitrary set B ⊆ A. For all of the operations f ∈ Σ (let n be the arity of f) let us look how f transformas the elements of Bn. It is not necessary that f(B) ⊆ 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 Bn 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 A2). 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(x1, ..., xi, ..., xn) ≡ xi). 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 x1, ..., xk−1, xk+1, ..., xn ∈ G the value of the expression f(x1, ..., xk−1, y, xk+1, ..., xn) is defined for not less that three different elements y ∈ 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.

1. Kozhukhov I. B., Reshetnikov A. V. 2010, “Algebras whose equivalence relations are congruences”, Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), vol. 16, no. 3, pp. 161–192. (Russian) translation in Journal of Mathematical Sciences (New York), 2011, Vol. 177, № 6, pp. 886–907

2. Gr¨atzer G. 2008, 2nd ed. with updates, 1979, Second Edition. “Universal algebra. Second Edition.”, Springer, Springer Science + Business Media. 586 p.

3. Ljapin E. S., Evseev A. E. 1991. “The Theory of Partial Algebraic Operations”, Obrazovanie, Herzen State Pedagogical University of Russia, St.Petersburg, 163 p. (Russian) translation in Springer, Springer Science + Business Media, B.V., 1997, 237 p.

4. Gr¨atzer G., Schmidt E. T. 1963, “Characterizations of congruence lattices of abstract algebras.”, Acta Sci. Math. (Szeged), vol.24, № 3. pp. 34–59.

5. G. Gr¨atzer, G. H. Wenzel. 1967, “On the concept of congruence relation in partial algebras.”, Math. Scand., vol. 20. pp. 275–280.

6. Reshetnikov A. V. 2011, “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), vol. 11, no. 3, part 2, pp. 46–51. (Russian)

7. Joel Berman. December 1971, “Strong congruence lattices of finite partial algebras”, Algebra Universalis, volume 1, issue 1, pp. 133–135.

8. Burmeister P. January 1970. “Free partial algebras.”, J. Reine Angew. Math., volume 1970, issue 241, pp. 75–86.

9. Clifford A. H., Hall T. E. 1973. “A characterisation of R-classes of semigroups as a partial groupoids.”, Semigroup Forum, volume 6, pp. 246–254.

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

11. Kulik V. T. 1974. “O reshetkah silnyh otnosheniy congruentnosti polugruppoidov.”, Uporyadochennye mnojestva i reshetki, no. 2. Saratov: izd. Saratovskogo Universiteta, pp. 42–50.

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

13. Pastijn F. 1976 “A generalization of Green’s equivalence relations for halfgroupoids.”, Simon Stevin, vol. 49, pp. 165–175.

14. Fleischer I. 1975. “On extending congruences from partial algebras.”, Fund. math., vol. 88, pp. 11–16.

15. Schelp R. H. 1972. “A partial semigroup approach to partially ordered sets.”, Proc. London Math. Soc., vol. 24, pp. 46–58.