Amit: A very interesting example of non-homogeneous posets are lotteries. Joel has done a significant amount of work on them.

Given a collection $\{{\mathbb P}_i\mid i\in I\}$ of posets, their lottery is $\{(p,i)\mid p\in {\mathbb P}_i\}$ ordered by $(p,i)\le(q,j)$ iff $i=j$ and $p \le_{{\mathbb P}_i} q$.

The generic "randomly" selects a ${\mathbb P}_i$ and adds a generic for it. By starting with wildly different posets, we may end up with radically different extensions, depending on the generic.

This is particularly useful in iterations, where along the way we need to ensure a variety of posets are selected (in order to obtain, say, certain forcing axioms). It is a nice alternative to using Laver functions.

The standard reference here is J. D. Hamkins, "The lottery preparation", Ann. Pure Appl. Logic 101 (2000), 103–146. For a (sophisticated) recent application, see for example N. Dobrinen - S. Friedman, "The consistency strength of the tree property at the double successor of a measurable cardinal", Fundamenta Mathematicae 208 (2010), 123–153. (In this paper, we start with a ground model where GCH holds and $\kappa$ is weakly compact hypermeasurable, and a poset is described that preserves the measurability of $\kappa$ while forcing the tree property at $\kappa^{++}$. The poset is an Easton support iteration of a lottery of iterated Sacks forcing posets at different cardinals.)

A good example of a poset that is as far from being homogeneous as you may want is the Vopenka algebra, see Theorem 15.46 of Jech's "Set Theory". Woodin realized that Vopenka algebras are particularly useful in the study of models of determinacy, and this has been a key insight.

In the original application, Vopenka showed that if $A$ a set of ordinals, then $L[A]$ is a generic extension of its HOD. Conditions are (ordinals coding) ordinal definable sets of subsets of $\kappa$, where $A\subseteq\kappa$. The usefulness of this algebra and its variants for determinacy is that it allows us to argue about arbitrary sets of reals as if they where Borel sets. For a concrete application, see for example my recent paper with Ketchersid, "A trichotomy theorem in natural models of AD${}^+$", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.

Amit: A very interesting example of non-homogeneous posets are lotteries. Joel introduced them and has done a significant amount of work on them.

Given a collection $\{{\mathbb P}_i\mid i\in I\}$ of posets, their lottery is $\{(p,i)\mid p\in {\mathbb P}_i\}$ ordered by $(p,i)\le(q,j)$ iff $i=j$ and $p \le_{{\mathbb P}_i} q$.

The generic "randomly" selects a ${\mathbb P}_i$ and adds a generic for it. By starting with wildly different posets, we may end up with radically different extensions, depending on the generic.

This is particularly useful in iterations, where along the way we need to ensure a variety of posets are selected (in order to obtain, say, certain forcing axioms). It is a nice alternative to using Laver functions.

The standard reference here is J. D. Hamkins, "The lottery preparation", Ann. Pure Appl. Logic 101 (2000), 103–146. For a (sophisticated) recent application, see for example N. Dobrinen - S. Friedman, "The consistency strength of the tree property at the double successor of a measurable cardinal", Fundamenta Mathematicae 208 (2010), 123–153. (In this paper, we start with a ground model where GCH holds and $\kappa$ is weakly compact hypermeasurable, and a poset is described that preserves the measurability of $\kappa$ while forcing the tree property at $\kappa^{++}$. The poset is an Easton support iteration of a lottery of iterated Sacks forcing posets at different cardinals.)

A good example of a poset that is as far from being homogeneous as you may want is the Vopenka algebra, see Theorem 15.46 of Jech's "Set Theory". Woodin realized that Vopenka algebras are particularly useful in the study of models of determinacy, and this has been a key insight.

In the original application, Vopenka showed that if $A$ a set of ordinals, then $L[A]$ is a generic extension of its HOD. Conditions are (ordinals coding) ordinal definable sets of subsets of $\kappa$, where $A\subseteq\kappa$. The usefulness of this algebra and its variants for determinacy is that it allows us to argue about arbitrary sets of reals as if they where Borel sets. For a concrete application, see for example my recent paper with Ketchersid, "A trichotomy theorem in natural models of AD${}^+$", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.

Amit: A very interesting example of non-homogeneous posets are lotteries. Joel has done a significant amount of work on them.

Given a collection $({\mathbb P}_i\mid i\in I)$ of posets, their lottery is $\{(p,i)\mid p\in {\mathbb P}_i\}$ ordered by $(p,i)\le(q,j)$ iff $i=j$ and $p\le_{{\mathbb P}_i}q$.

The generic "randomly" selects a ${\mathbb P}_i$ and adds a generic for it. By starting with wildly different posets, we may end up with radically different extensions, depending on the generic.

This is particularly useful in iterations, where along the way we need to ensure a variety of posets are selected (in order to obtain, say, certain forcing axioms). It is a nice alternative to using Laver functions.

The standard reference here is J. D. Hamkins, "The lottery preparation", Ann. Pure Appl. Logic 101 (2000), 103–146. For a (sophisticated) recent application, see for example N. Dobrinen - S. Friedman, "The consistency strength of the tree property at the double successor of a measurable cardinal", Fundamenta Mathematicae 208 (2010), 123–153. (In this paper, we start with a ground model where GCH holds and $\kappa$ is weakly compact hypermeasurable, and a poset is described that preserves the measurability of $\kappa$ while forcing the tree property at $\kappa^{++}$. The poset is an Easton support iteration of a lottery of iterated Sacks forcing posets at different cardinals.)

A good example of a poset that is as far from being homogeneous as you may want is the Vopenka algebra, see Theorem 15.46 of Jech's "Set Theory". Woodin realized that Vopenka algebras are particularly useful in the study of models of determinacy, and this has been a key insight.

In the original application, Vopenka showed that if $A$ a set of ordinals, then $L[A]$ is a generic extension of its HOD. Conditions are (ordinals coding) ordinal definable sets of subsets of $\kappa$, where $A\subseteq\kappa$. The usefulness of this algebra and its variants for determinacy is that it allows us to argue about arbitrary sets of reals as if they where Borel sets. For a concrete application, see for example my recent paper with Ketchersid, "A trichotomy theorem in natural models of AD${}^+$", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.

Amit: A very interesting example of non-homogeneous posets are lotteries. Joel has done a significant amount of work on them.

Given a collection $\{{\mathbb P}_i\mid i\in I\}$ of posets, their lottery is $\{(p,i)\mid p\in {\mathbb P}_i\}$ ordered by $(p,i)\le(q,j)$ iff $i=j$ and $p \le_{{\mathbb P}_i} q$.

The generic "randomly" selects a ${\mathbb P}_i$ and adds a generic for it. By starting with wildly different posets, we may end up with radically different extensions, depending on the generic.

This is particularly useful in iterations, where along the way we need to ensure a variety of posets are selected (in order to obtain, say, certain forcing axioms). It is a nice alternative to using Laver functions.

The standard reference here is J. D. Hamkins, "The lottery preparation", Ann. Pure Appl. Logic 101 (2000), 103–146. For a (sophisticated) recent application, see for example N. Dobrinen - S. Friedman, "The consistency strength of the tree property at the double successor of a measurable cardinal", Fundamenta Mathematicae 208 (2010), 123–153. (In this paper, we start with a ground model where GCH holds and $\kappa$ is weakly compact hypermeasurable, and a poset is described that preserves the measurability of $\kappa$ while forcing the tree property at $\kappa^{++}$. The poset is an Easton support iteration of a lottery of iterated Sacks forcing posets at different cardinals.)

A good example of a poset that is as far from being homogeneous as you may want is the Vopenka algebra, see Theorem 15.46 of Jech's "Set Theory". Woodin realized that Vopenka algebras are particularly useful in the study of models of determinacy, and this has been a key insight.

In the original application, Vopenka showed that if $A$ a set of ordinals, then $L[A]$ is a generic extension of its HOD. Conditions are (ordinals coding) ordinal definable sets of subsets of $\kappa$, where $A\subseteq\kappa$. The usefulness of this algebra and its variants for determinacy is that it allows us to argue about arbitrary sets of reals as if they where Borel sets. For a concrete application, see for example my recent paper with Ketchersid, "A trichotomy theorem in natural models of AD${}^+$", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258.