CAT algebra

consider the unital commutative algebra defined by

c^2 = a^2 = t^2 = -1

ca = ac = t

ct= = tc = a

at = ta = c

Every number is a square, in other words :

For every given y there exists at least one solution x such that

x^2 = y

In fact we have a solution x^3 = 0 for nonzero x ;

(c + a)^3 = (t + a)^3 = (c + t)^3 = 0

Notice there is NO solution other than 0 to the equation

x^2 = 0

This algebra is not associative !

Thank you for your interest.

tommy1729

update : generalizations are under research and might be uploaded !