The identity: empty string  
The operation: concatenation

("dog" ++ "cat") ++ "pig" = "dog" ++ ("cat" ++ "pig")  
"" ++ "dog" = "dog" ++ "" = "dog"

The identity: 0  
The operation: Add 

1 + 2 + 3 = 1 + (2 + 3)
0 + 1 = 1 + 0 = 1

The identity: Mininum, Maximum  
The operation: Maximum( , ), Minimum( , ) 

Max(Max(a, b), c) = Max(a, Max(b, c))
Max(Min, a) = a

Min(Min(a, b), c) = Min(a, Min(b, c))
Min(Max, a) = a

Identity: True
Operation: Or

Identity: False
Operation: And

True || False = False || True = False
False && True = True && False = False

Identity: [], empty list
Operation: length

length [] + length [1] = length [1] + length []
length [1, 2] + length [3, 4] + length [5] = length [1, 2] + (length [3, 4] + length [5]) 

Identity: []
Operation: merge 

(+++:)::(Ord a)=>[a]->[a]->[a]
(+++:) xs [] = xs 
(+++:) [] ys = ys 
(+++:) (x:xs) (y:ys) = if x <= y then x:(xs +++: (y:ys)) else y:((x:xs) +++: ys)

main = do 
        pp $ [] +++: [1] == [1]  -- left identity
        pp $ [2] +++: [] == [2]  -- right identity
        pp $ [2] +++: [1] == [1, 2] -- a, b \in S, then a `op` b \in S  
        pp $ [1, 2] +++: [0, 1, 9, 10] == [0, 1, 1, 2, 9, 10] 
        -- associativity
        pp $ ([1, 2] +++: [0, 1] +++: [4, 5]) == ([1, 2] +++: ([0, 1] +++: [4, 5]))