Definition. Let M be a set. By a binary operation on M we mean any - PDF document

Definition. Let M be a set. By a binary operation on M we mean any mapping ◦ : M × M �→ M .

Definition. Consider a binary operation ◦ on a set M . We say that the couple ( M, ◦ ) is a monoid if it satisfies the following axioms: (A1) associative law : x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ z for all x, y, z ∈ M . (A2) existence of an identity element : There is e ∈ M such that x ◦ e = e ◦ x = x for all x ∈ M .

Fact. Let ◦ be a binary operation on a set M . If there are identity elements e, f in ( M, ◦ ), then e = f .

Definition. Let ( M, ◦ , e ) be a monoid. We say that an element x ∈ M is invertible if there is y ∈ M such that x ◦ y = y ◦ x = e .

Fact. Let ( M, ◦ , e ) be a monoid, let x ∈ M be invertible. Assume that y 1 , y 2 ∈ M satisfy x ◦ y 1 = y 1 ◦ x = e and x ◦ y 2 = y 2 ◦ x = e . Then y 1 = y 2 .

Definition. Let ( M, ◦ , e ) be a monoid and x ∈ M be an invertible element. The element y ∈ M satisfying x ◦ y = y ◦ x = e is called the inverse of x and we denote it x − 1 .

( Z 4 , ⊕ , 0): ⊕ 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 ( Z 14 , ⊙ , 1): ⊙ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 3 0 3 6 9 12 1 4 7 10 13 2 5 8 11 4 0 4 8 12 2 6 10 0 4 8 12 2 6 10 5 0 5 10 1 6 11 2 7 12 3 8 13 4 9 6 0 6 12 4 10 2 8 0 6 12 4 10 2 8 7 0 7 0 7 0 7 0 7 0 7 0 7 0 7 8 0 8 2 10 4 12 6 0 8 2 10 4 12 6 9 0 9 4 13 8 3 12 7 2 11 6 1 10 5 10 0 10 6 2 12 8 4 0 10 6 2 12 8 4 11 0 11 8 5 2 13 10 7 4 1 12 9 6 3 12 0 12 10 8 6 4 2 0 12 10 8 6 4 2 13 0 13 12 11 10 9 8 7 6 5 4 3 2 1

Theorem. Let ( M, ◦ , e ) be a monoid. (i) If x ∈ M is invertible, then also its inverse x − 1 is invertible and ( x − 1 ) − 1 = x . (ii) If x, y ∈ M are invertible, then also x ◦ y is invertible and ( x ◦ y ) − 1 = y − 1 ◦ x − 1 .

Fact. Let ( M, ◦ , e ) be a monoid. Asume that elements c, x, y ∈ M satisfy equality c ◦ x = c ◦ y or equality x ◦ c = y ◦ c . If c is invertible then x = y .

Definition. Consider a binary operation ◦ on a set M . We say that the couple ( A, ◦ ) is a group if it satisfies the following conditions: (A1) asociative law : x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ z for all x, y, z ∈ M . (A2) existence of an identity element : There is e ∈ M such that x ◦ e = e ◦ x = x for all x ∈ M . (A3) existence of inverse elements : For every x ∈ M there is y ∈ M such that x ◦ y = y ◦ x = e .

Definition. Let ( M, ◦ ) be a monoid. Pro any n ∈ N and x ∈ M we define the powers of x as follows: (0) x 1 = x ; (1) x n +1 = ( x n ) ◦ x for n ∈ N . If x is invertible, then for n ∈ N we also define x − n = ( x − 1 ) n . Convention: Power has priority over the operation ◦ , unless paren- theses say otherwise.

Fact. Let ( M, ◦ ) be a monoid and let x ∈ M . Then for all m, n ∈ N the following are true: (i) x m ◦ x n = x m + n , (ii) ( x m ) n = x mn . Moreover, if x is invertible, then these identities are true for all m, n ∈ Z .

Fact. Let ( M, ◦ , e ) be a monoid, let g � = e be an invertible element of M . Assume that the set A = { n ∈ N ; ∃ k ∈ { 1 , 2 , . . . , n − 1 } : g n = g k } is not empty, denote its least element as m . Then g m = g . Also g m − 1 = g m ◦ g − 1 = g ◦ g − 1 = e .

Definition. Let ( M, ◦ , e ) be a monoid and let g ∈ M be invertible. We define the order of the element g , denoted ord( g ), as the least element of the set { n ∈ N ; g n = e } in case when it is not empty, otherwise we define ord( g ) = ∞ .

Definition. Let g be some invertible element in a monoid ( M, ◦ , e ). We define the group generated by g as � g � = { g n ; n ∈ Z } .

Theorem. Let ( G, ◦ , e ) be a group, where G is a finite set. Then ord( g ) divides | G | for every g ∈ G . Therefore also g | G | = e .

Theorem. Let ( M, ◦ , e ) be a monoid. Denote N = { x ∈ M ; x invertible } . Then ( N, ◦ , e ) is a group.

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Definition. We define the Euler function or totient ϕ as follows: For n ∈ N we set ϕ ( n ) to be the number of natural numbers that are less than n and coprime with n .

Theorem. (Euler’s theorem) Let n ∈ N . If a ∈ N is coprime with n , then a ϕ ( n ) ≡ 1 (mod n ).

Theorem. Let m, n ∈ N be coprime. Then ϕ ( m · n ) = ϕ ( m ) · ϕ ( n ).