**Product Forcing**

Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.

**Finite products**: If*P*and*Q*are posets, the product poset*P*×*Q*has the partial order defined by (*p*_{1},*q*_{1}) ≤ (*p*_{2},*q*_{2}) if*p*_{1}≤*p*_{2}and*q*_{1}≤*q*_{2}.**Infinite products**: The product of a set of posets*P*_{i},*i**I*, each with a largest element 1 is the set of functions*p*on*I*with*p*(*i*)*P*(*i*) and such that*p*(*i*) = 1 for all but a finite number of*i*. The order is given by*p*≤*q*if*p*(*i*) ≤*q*(*i*) for all*i*.- The
**Easton product**(after William Bigelow Easton) of a set of posets*P*_{i},*i**I*, where*I*is a set of cardinals is the set of functions*p*on*I*with*p*(*i*)*P*(*i*) and such that for every regular cardinal γ the number of elements α of γ with*p*(α) ≠ 1 is less than γ.

Read more about this topic: List Of Forcing Notions

### Famous quotes containing the words forcing and/or product:

“It is quite true, as some poets said, that the God who created man must have had a sinister sense of humor, creating him a reasonable being, yet *forcing* him to take this ridiculous posture, and driving him with blind craving for this ridiculous performance.”

—D.H. (David Herbert)

“Evil is committed without effort, naturally, fatally; goodness is always the *product* of some art.”

—Charles Baudelaire (1821–1867)