We construct an algebra A such that A has a nonempty finite set Δ of associative and commutative binary operations. Then we may define an ideal with respect to a nonempty subset of Δ. If some hypotheses are satisfied, then we have that a union of the ideals is an ideal. An ideal M is maximal with respect to a subset of Δ if there is not an ideal J ≠ A such that J contains M. And an algebra is local with respect to a subset of Δ if it has a unique maximal ideal. Suppose that the algebra A is local with respect to Φ and Ψ, M and N are the maximal ideals, respectively, and J is an ideal with respect to Φ∪Ψ. Then we have that J ⊆ M∩ N if some conditions hold. Let A be a local algebra with respect to Φ, M the maximal ideal. For all Ψ with Φ ⊆ Ψ⊂ Δ, if M is an ideal with respect to Ψ, then A is local with respect to Ψ. A preimage of an ideal with respect to Φ under a homomorphism is an ideal with respect to Φ.