CHARACTERIZATION OF POINTED VARIETIES OF UNIVERSAL ALGEBRAS WITH NORMAL PROJECTIONS
Zurab Janelidze · 2003
We characterize pointed varieties of universal algebras in which (A × B)/A ≈ B, i.e. all product projections are normal epimorphisms. 1. Definition. We will say that a pointed category C has normal projections if every product projection A × B → B in C is a normal epimorphism. Equivalently, for any two objects A and B in such a category C, forming the product A×B and then factoring it by A ≈ A× 0r esults inB. In particular, every Jonsson-Tarski variety of universal algebras (3) (considered as a category) has this property; the same is true for the pointed subtractive varieties in the sense of Ursini (4). The purpose of this paper is to characterize pointed varieties with normal projections (Theorem 3 below). Before stating the theorem, we make a simple reformulation of Definition 1. 2. Proposition. Let C be a pointed variety. The following conditions are equivalent: (a) C has normal projections; (b) there exists a natural number n, such that for all A and B in C ,a nd for alla ∈ A and b ∈ B, ((a,b),(0,b)) ∈ R n ,w hereR is the reflexive homomorphic relation on A × B generated by the set {((a,0),(a,0))| a =0 or a =0 }; (c) let F(x) be the free algebra in C generated by x; there exists a natural number n such that ((x,x),(0,x)) ∈ Q n ,w hereQ is the reflexive homomorphic relation on F(x) × F(x) generated by the set {((x,0),(0,0)),((0,0),(x,0))}. Moreover, the number n in (b) and in (c) can be supposed to be the same.