Normal subalgebras. I (Q354676)

Let \(\mathcal V\) be a variety of (universal) algebras. A subalgebra of \(A\in \mathcal V\) is \textit{normal} if it is the universe image under some morphism of the subalgebra generated by constants in the target. Consider \(\mathbb{C}_{\mathcal V}\) (or just \(\mathbb{C}\)), the free algebra in \(\mathcal V\) over the empty set (the initial algebra in the category \(\mathcal V\)) and let \(E\subseteq \mathbb{C}_{\mathcal V}\). Assume \(E\neq \emptyset\). The author extends the notions of normal subalgebras, clots and ideals of \(A\) and introduces the notions: \(E\)-complex, \(E\)-normal complex, \(E\)-normal subalgebra, \(E\)-clot and \(E\)-ideal of \(A\). He defines the notions of \(E\)-subtractive varieties, \(E\)-permutable congruence and the coherence of an algebra, too. \(E\)-subtractivity has some connections with the property of coherence for congruences and some consequences. For instance, it allows a simple description of the join of two \(E\)-ideals and implies that the lattice of \(E\)-ideals is modular. Also, the author presents characterizations and connections of all the notions mentioned above and studies some special cases (in the varieties of Heyting algebras, Boolean algebras and unitary rings). He proves that normal \(E\)-complexes of an algebra \(A\) are \(E\)-clots of \(A\); \(E\)-clots of algebras in \(\mathcal V\) have in turn another algebraic property which makes them into \(E\)-ideals (relative to \(\mathcal V\)). The paper ends with a short presentation of the second part of it.