Normal Subalgebras, I

Abstract

We extend the notions of normal subalgebras, clots and ideals of an algebra A in a variety of (universal) algebras, from the familiar case of a single constant to the case of any number of constants. The first idea is that a subalgebra of A is normal when it is the inverse image under some morphism of the subalgebra generated by constants in the target. We argue that a better approach is obtained by considering pullbacks of γB and g : A → B, where g : A → B is some morphism and γB is the morphism from the initial algebra of the variety to B. Examples are shown in Heyting algebras, boolean algebras and unitary rings. Ideals and clots are generalizations of this notion, defined instead by closure under derived operations which have the right behavior on constants. There are several characterizations of these notions; some of them aiming at a categorical generalization. We deal with an (extended) notion of subtractivity, showing that it implies that ideals coincide with normal subalgebras, and it is connected with notions of coherence of congruences, allowing a characterization of protomodular varieties.