Definition: Let A = (S,,',...........) be an ordered pair where S is a nonempty set and (exponent): S X S ---->S is a binary operation on S. Then we call (S,,*',...........) an algebraic structure.
Clearly,this gives a general definition which allows us to define a category AlgStruc i.e. the category of all algebraic structures. Now let's state a simple proposition:
Let 0 in A be a zero element (i.e. * on S for A is ordinary addition and for every x in S, x+0=0+x=x) and let 1 in A be the 1 element ( i.e. ' on S where for every x in S, x'1=1*'x=x). Then the following "subcategory" is empty in AlgStruc: { A ! 0 =1 for every x in S} .
In other words,there is no algebraic structure with at least one binary operation defined on it where the zero element and the 1 element are the same. This is FALSE;consider the trivial semigroup ({0},*) Clearly 0=1 in this case!
