Ideal in a variety with zero
Let be a variety of algebras with zero. In other words, 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 is an algebra in . An ideal in is a nonempty subset , with the following property:
For any expression constructed using the operators of the operator domain, such that whenever all the s are zero, takes the value zero, it is true that when all the are in , takes a value inside .
Such expressions are termed ideal terms.
Relation with other properties
- 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.