Ideal in a variety with zero

Definition
Let $$\mathcal{V}$$ be a variety of algebras with zero. In other words, $$\mathcal{V}$$ has an operator domain comprising operators with various arities, some universal identities satisfied by these operators, and a distinguished constant operator among these, called the zero operator.

Suppose $$A$$ is an algebra in $$\mathcal{V}$$. An ideal in $$A$$ is a nonempty subset $$S$$, with the following property:

For any expression $$\varphi(u_1,u_2,\dots,u_m,t_1,t_2,\dots,t_n)$$ constructed using the operators of the operator domain, such that whenever all the $$u_j$$s are zero, $$\varphi$$ takes the value zero, it is true that when all the $$u_j$$ are in $$S$$, $$\varphi$$ takes a value inside $$S$$.

Such expressions are termed ideal terms.

Stronger properties

 * Weaker than::Kernel of a congruence: The kernel of a congruence is defined as the inverse image of zero under the quotient map arising from the congruence. The kernel of any congruence must be an ideal. This gives a natural map from the collection of all congruences to the collection of all ideals, which need not in general be either injective or surjective. When the map is a bijection, we say that the variety is ideal-determined. The variety of groups is ideal-determined.

Weaker properties

 * Stronger than::Clot