Semigroups of polynomials

Definitions. A set S of polynomials in x with integer coefficients is called a semigroup of polynomials if the composition of any two elements of S is an element of S, that is, S is closed under compostion. The semigroup is a commuative semigroup if each two polynomials in S commute with each other.

We are interested in finding semigroups of polynomials.

Problem. Show that the constant polynomials form a semigroup of polynomials. Is it commutative?

Problem. Show that the linear polynomials form a semigroup of polynomials. Is it commutative?

Problem. Show that the linear polynomials of slope 1 form a semigroup of polynomials. Is it commutative?

Problem. Show that the linear polynomials with y-intercep 0 form a semigroup of polynomials. Is it commutative?

Definition. A set S with an associative binary operation * is called a semigroup .

Problem. Are semigroups of polynomials semigroups? Why?

Definition Two semigroups S and T are isomorphic if there is a 1-1 function f from S onto T which preserves the operation in the sense that f(x*y) = f(x)*f(y). Such a function f is called an isomorphism from S to T.

Problem. Show that the semigroup of polynomials of slope 1 is isomorphic with the semigroup of integers under the operation of addition.

Problem. Let S be the set of polynomials as n runs over the integers. Show this is a semigroup of polynomials. What familiar semigroup is isomorphic with S?

Definition. A set T of matrices is called a semigroup of matrices if it is closed under matrix multiplication.

Problem. Let S be the set of linear polynomials , a and b integers. Let T be the set of matrices , a and b integers. Show that T is a semigroup of matrices and that the function f from S to T defined by f( ) = is an isomorphism.

Problem. Show that the polynomials , form a semigroup of polynomials.

Problem. Let , , and for n > 1, . Plot the first few of these over the interval -1 to 1. Investigate whether these polynomials commute with each other.