For each property P of kind K there is a property p of kind k such that it is a law that: for all things x and times t: x has P at t iff x has p at t.
(These are also called bridge laws, hence the name of this kind of reduction.)
Suppose also that we can derive all the laws about properties of kind K from the laws about properties of kind k by using these lawful correlations.
Then there is a sense in which properties of kind K reduce to properties of kind k. This is bridge-law reduction.
Then we can derive the law about mental properties from the law about brain properties using the two lawful correlations, and mental properties bridge-law reduce to brain properties.
For each property P of kind K there is a property p of kind k such that P = p.
Then there is a sense in which properties of kind K reduce to properties of kind k. This is identity reduction.
(Although Kim thinks it denies that there is any correlation to explain.)
For each property P of kind K there is a role C such that to have P is to have some property p that plays role C.
Suppose also that for each property P of kind K the role associated with P is played by a property of kind k.
Then there is a sense in which properties of kind K reduce to properties of kind k. This is functional reduction.
Transparency might be like this. It doesn't make sense to talk about the transparency of a single water molecule, or of a small number of water molecules - the property doesn't really exist at this level. But gather together sufficiently many water molecules and it does start to make sense - the property exists at this higher, more complex level.
This is unlike the property of having mass, which is sometimes called a resultant or additive property.
We might employ the following slogan to capture the idea of emergence: the whole can be more than the some of the parts.