This first edition was written for Lua 5.0. While still largely relevant for later versions, there are some differences.
The third edition targets Lua 5.2 and is available at Amazon and other bookstores.
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16.2 – Inheritance

Because classes are objects, they can get methods from other classes, too. That makes inheritance (in the usual object-oriented meaning) quite easy to implement in Lua.

Let us assume we have a base class like Account:

    Account = {balance = 0}
    
    function Account:new (o)
      o = o or {}
      setmetatable(o, self)
      self.__index = self
      return o
    end
    
    function Account:deposit (v)
      self.balance = self.balance + v
    end
    
    function Account:withdraw (v)
      if v > self.balance then error"insufficient funds" end
      self.balance = self.balance - v
    end

From that class, we want to derive a subclass SpecialAccount, which allows the customer to withdraw more than his balance. We start with an empty class that simply inherits all its operations from its base class:

    SpecialAccount = Account:new()
Up to now, SpecialAccount is just an instance of Account. The nice thing happens now:
    s = SpecialAccount:new{limit=1000.00}
SpecialAccount inherits new from Account like any other method. This time, however, when new executes, the self parameter will refer to SpecialAccount. Therefore, the metatable of s will be SpecialAccount, whose value at index __index is also SpecialAccount. So, s inherits from SpecialAccount, which inherits from Account. When we evaluate
    s:deposit(100.00)
Lua cannot find a deposit field in s, so it looks into SpecialAccount; it cannot find a deposit field there, too, so it looks into Account and there it finds the original implementation for a deposit.

What makes a SpecialAccount special is that it can redefine any method inherited from its superclass. All we have to do is to write the new method:

    function SpecialAccount:withdraw (v)
      if v - self.balance >= self:getLimit() then
        error"insufficient funds"
      end
      self.balance = self.balance - v
    end
    
    function SpecialAccount:getLimit ()
      return self.limit or 0
    end
Now, when we call s:withdraw(200.00), Lua does not go to Account, because it finds the new withdraw method in SpecialAccount first. Because s.limit is 1000.00 (remember that we set this field when we created s), the program does the withdrawal, leaving s with a negative balance.

An interesting aspect of OO in Lua is that you do not need to create a new class to specify a new behavior. If only a single object needs a specific behavior, you can implement that directly in the object. For instance, if the account s represents some special client whose limit is always 10% of her balance, you can modify only this single account:

    function s:getLimit ()
      return self.balance * 0.10
    end
After that declaration, the call s:withdraw(200.00) runs the withdraw method from SpecialAccount, but when that method calls self:getLimit, it is this last definition that it invokes.