MathDB

A set

M

of elements

u, v, w

is called a semigroup if an operation is defined in it is which uniquely assigns an element

w

from

M

to every ordered pair

(u, v)

of elements from

M

(you write

u \otimes v = w

) and if this algebraic operation is associative, i.e. if for all elements

u, v,w

from

M

(u \otimes v) \otimes w = u \otimes (v \otimes w).

Now let

c

be a positive real number and let

M

be the set of all non-negative real numbers that are smaller than

c

. For each two numbers

u, v

from

M

we define:

u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}

Investigate a) whether

M

is a semigroup; b) whether this semigroup is regular, i.e. whether from

u \otimes v_1 = u\otimes v_2

always

v_1 = v_2

and from

v_1 \otimes u = v_2 \otimes u

also

v_1 = v_2

follows.

Problem Statement