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 y1, y2 ∈ M satisfy x ◦ y1 = y1 ◦ x = e and x ◦ y2 = y2 ◦ x = e. Then y1 = y2.

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.

(Z4, ⊕, 0): ⊕ 0 1 2 3 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 (Z14, ⊙, 1): ⊙ 1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 2 3 4 5 6 7 8 9 10 11 12 13 2 2 4 6 8 10 12 0 2 4 6 8 10 12 3 3 6 9 12 1 4 7 10 13 2 5 8 11 4 4 8 12 2 6 10 0 4 8 12 2 6 10 5 5 10 1 6 11 2 7 12 3 8 13 4 9 6 6 12 4 10 2 8 6 12 4 10 2 8 7 7 7 7 7 7 7 7 8 8 2 10 4 12 6 8 2 10 4 12 6 9 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) x1 = x; (1) xn+1 = (xn) ◦ 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) xm ◦ xn = xm+n, (ii) (xm)n = xmn. 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}: gn = gk} is not empty, denote its least element as m. Then gm = g. Also gm−1 = gm ◦ 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; gn = e} in case when it is not empty,

• therwise 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 = {gn; 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 aϕ(n) ≡ 1 (mod n).

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