MathDB

Define the operator "

*

" on

\mathbb{R}

x*y=x+y+xy.

a) Show that

\mathbb{R}\setminus\{ -1\} ,

along with the operator above, is isomorphic with

\mathbb{R}\setminus\{ 0\} ,

with the usual multiplication.
b) Determine all finite semigroups of

\mathbb{R}

under "

*.

" Which of them are groups?
c) Prove that if

H\subset_{*}\mathbb{R}

is a bounded semigroup, then

H\subset [-2, 0].

Problem Statement