Query to get the highest rank of each item

Question

I've the(simplified) following model:

Book
  id
  name

BookCategory
  book_id
  category_id
  rank

Category
  id
  name

With a given category id, I'd like to get the books having that category as the highest ranked one.

I'll give an example to be more clear about it:

Book   

id    name   
---  -------
1     On Writing
2     Zen teachings
3     Siddharta

BookCategory   

book_id category_id      rank   
---       -------        -----
1        2               34.32
1        5               24.23
1        9               54.65
2        5               27.33
2        9               28.32
3        2               30.43
3        5               27.87

Category   

id    name   
---  -------
2     Writing
5     Spiritual
9     Buddism

The result for category_id = 2 would be the book with id = 3.

This is the query I'm running:

SELECT book."name" AS bookname
FROM bookcategory AS bookcat
LEFT JOIN book ON bookcat."book_id" = book."id" 
LEFT JOIN category cat ON bookcat."category_id" = cat."id" 
WHERE cat."id" = 2
ORDER BY bookcat."rank"

This is not the right way to do it because it doesn't select the max rank of each book. I've yet to find a proper solution.

Note: I'm using the postgresql 9.1 version.

Edit:

DB Schema (taken from martin's SQL Fiddle answer):

create table Book (
  id int,
  name varchar(16)
  );

insert into Book values(1, 'On Writing');
insert into Book values(2, 'Zen teachings');
insert into Book values(3, 'Siddharta');

create table BookCategory (
  book_id int,
  category_id int,
  rank real
  );

insert into BookCategory values(1,2,34.32);
insert into BookCategory values(1,5,24.23);
insert into BookCategory values(1,9,54.65);
insert into BookCategory values(2,5,27.33);
insert into BookCategory values(2,9,28.32);
insert into BookCategory values(3,2,30.43);
insert into BookCategory values(3,5,27.87);

create table Category (
  id int,
  name varchar(16)
  );

insert into Category values(2, 'Writing');
insert into Category values(5,'Spiritual');
insert into Category values(9,    'Buddism');

Answer 1

add another column to calculate rank:

dense_rank() OVER (PARTITION BY book."name" ORDER BY bookcat."rank"
s ASC) AS rank

Answer 2

To set up:

CREATE TABLE Book
(
  id int PRIMARY KEY,
  name text not null
);

CREATE TABLE Category
(
  id int PRIMARY KEY,
  name text not null
);

CREATE TABLE BookCategory
(
  book_id int,
  category_id int,
  rank numeric not null,
  primary key (book_id, category_id)
);

INSERT INTO Book VALUES
  (1, 'On Writing'),
  (2, 'Zen teachings'),
  (3, 'Siddharta');

INSERT INTO Category VALUES
  (2, 'Writing'),
  (5, 'Spiritual'),
  (9, 'Buddism');

INSERT INTO BookCategory VALUES
  (1, 2, 34.32),
  (1, 5, 24.23),
  (1, 9, 54.65),
  (2, 5, 27.33),
  (2, 9, 28.32),
  (3, 2, 30.43),
  (3, 5, 27.87);

The solution:

SELECT Book.name
  FROM (
         SELECT DISTINCT ON (book_id)
             *
           FROM BookCategory
           ORDER BY book_id, rank DESC
       ) t
  JOIN Book ON Book.id = t.book_id
  WHERE t.category_id = 2
  ORDER BY t.rank;

Logically, the subquery in the FROM clause generates a relation with the highest ranking category for each book, from which you then select the books in that category and order them by the ranking in that category.

Results:

   name    
-----------
 Siddharta
(1 row)