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.

Relation with other properties

Stronger 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.

Weaker properties

• Clot