Ideal in a variety with zero

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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 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_js 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.