# Ideal in a variety with zero

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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 subset $S$ containing zero, 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 $S$.