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 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_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.

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