Not really. You can see the basic idea on the nLab: https://ncatlab.org/nlab/show/variety+of+algebras#free_algeb...

Continuing the example I gave, the monad law "Identity" says that "trivial subexpressions are already simplified"

     a+b  = (a)+(b)
     ||       ||
    (a+b)  =  a+b

and the monad law "Associativity" says that "order of simplification doesn't matter"

    ((a)+(b))+((c+d)+(e)) = (a)+(b)+(c+d)+(e)
           ||                    ||
      (a+b)+(c+d+e)     =     a+b+c+d+e

The laws for an algebra for a monad say "Trivial expressions evaluate to themselves"

    a = (a) = a

and "you get the same result no matter the order you do evaluations and simplifications"

    (6+1)+(3) = 7+3 
       ||        ||
      6+1+3   =  10