In this paper we determine the smallest equivalence relation on a multialgebra for which the factor multialgebra is a universal algebra satisfying a given identity. We also establish an important property for the factor multialgebra (of a multialgebra) modulo this relation.2000 Mathematics subject classification: primary 08A99; secondary 08A30, 20N20.

Based on the properties of the poset of those equivalence relations of a multialgebra for which the factor multialgebra is a universal algebra, we give a characterization for the fundamental relations of a multialgebra. We point out the benefits of our approach by giving two applications. One of them provides a new characterization of the commutative fundamental relation of a hyperring, and the other will give a general category theoretical property of the construction of the fundamental algebras (both in the general case and in the hyperring case).

Our general investigation of universal algebras obtained from multialgebras via strongly regular equivalence relations provides useful general results concerning fuzzy set topics related to multialgebra theory. We also give many hints on how to connect our approach with the results from the literature.

No abstract

Let r be a hereditary torsion theory in ft-Mod. Then any ring homomorphism 7 : R -> 5 induces in S-Mod a torsion theory a given by the condition that a left 5-module is cr-torsion if and only if it is r-torsion as a left il-module. We show that if 7 : R -> S is a ring epimorphism and A is a r-injective left .R-module, then Ann/i Ker (7) is cr-injective as a left 5-module. As a consequence, we relate r-injectivity and cr-injectivity, and we give some applications.