Representing the anova's interaction in R I read in the book The basics of S and S-plus that "A:B" functions as "Interaction between A and B".
However, in other book I read "There is never an interaction effect for two factors when one is nested within the other".
But aov(y~a/b) would have the same results  as aov(y~a+a:b).  
So it seems to me a:b could mean interaction, but also could mean the effect of a factor nested within another. In other words, A:B" definition as "Interaction between A and B" is incomplete.
Am I right? How to explain all this?
 A: Be very careful with : it means a bunch of different things depending on the context with which you use it in R. 
See and ?interaction, ?formula, ?lm, ?':'
Here's an example of interaction: 
df <- data.frame(X=sample(letters[1:10],200, replace=T),Y=sample(letters[1:10],200, replace=T))
> df$X:df$Y
  [1] a:a i:i c:e g:g e:c e:h i:i j:f h:f a:j i:e c:h h:c h:a j:f i:g g:e a:c d:g f:j c:i h:g g:h g:d e:b a:a c:a
 [28] e:e c:d b:e i:h i:j g:g d:b h:d j:d a:j e:i d:g i:e e:c e:e h:h j:b f:b a:g h:g b:j h:e j:d b:f d:i j:i b:c
 [55] a:i c:b b:d g:h g:f h:i e:a h:e d:e d:f i:j a:a d:e i:b g:c d:g j:h c:g j:b i:d b:g e:c h:b e:g b:b h:g d:j
 [82] j:i i:b d:a a:h h:f j:c c:j f:j e:g h:i g:f j:a b:e j:i a:j d:c g:j a:h h:c b:a c:f b:e f:d c:d j:d i:f d:j
[109] g:b j:i c:c h:b b:a f:c c:g j:i h:b j:e c:j c:b i:e f:i c:j g:i i:e h:i b:e i:d c:i j:i h:g g:j d:j a:h d:b
[136] c:f j:b a:e f:i c:j j:h g:i b:d i:j h:i g:i g:i g:j h:d g:g g:c f:g e:b j:a b:b f:e i:i g:h c:f i:f f:c a:f
[163] h:g e:f b:b b:j b:a i:g b:i h:j f:j a:f h:j b:a f:d h:g f:f d:a d:j d:g d:g g:b d:e e:b h:b g:a h:a h:g e:j
[190] d:d d:b e:h h:j f:g a:g f:i j:b d:h a:g j:e
100 Levels: a:a a:b a:c a:d a:e a:f a:g a:h a:i a:j b:a b:b b:c b:d b:e b:f b:g b:h b:i b:j c:a c:b c:c ... j:j

Essentially returning all the unique set of X that match Y
In aov, or lm, or and function that takes a formula as an arguemnt. It means the "interaction effect" of those two variables. 
df <- data.frame(y=runif(100),x=rnorm(100),z=rchisq(100,20))
lm(y ~ x + z, df)
lm(y ~ x:z, df) 
lm(y ~ x*z, df) ## essentailly y~ x + z + x:z

It can also be short hand when using numbers, or objects that store numbers: 
1:4 = 1,2,3,4
a <- 5
b <- 10
a:b 5,6,7,8,9,10

