Abstract Algebra

7 flashcards · Mathematics · by lakesare

1. What's a groupoid?

   Nonempty set S Binary operation ○ Example of a grupoid: S is the whole numbers, and ○ is defined by a ○ b = a + 5b.7 ○ 8 = 7 + 5*8 = 47.This particular grupoid is not associative. (Binary operation is a function domain of which is a set of 2-tuples).

2. What's a semigroup?

   Grupoid that's associative.

3. What's a monoid?

   Semigroup that has an identity element.

4. What's a group?

   Monoid in which every element has an inverse. E.g., the inverse of (2, 3, 9) vector is (-2, -3, -9)

5. What's an abelian group?

   Group that is commutative. a ○ b = b ○ a for every a and b in S.

6. What's a ring?

   Nonempty set S, and two binary operations ○ and ●.(S, ○, ●) is called a ring when: If (S, ●) is an abelian group (S, ●) is a semigroup left distributive law holds a ● (b ● c) = (a ● b) ● (a ● c) right distributive law holds (a ● b) ● c = (a ● c) ● (b ● c)

7. What's inverse with respect to (ℝ, +) additive inverse for real numbersinverse with respect to (ℝ, *) multiplicative inverse for real numbers

   a ○ b = i where i - identity element for this (S, ○) inverse with respect to (ℝ, +): -x The identity element for (ℝ, +) is 0: a + 0 = 0 + a = aFor real number x the inverse is -x, because x + (-x) = 0inverse with respect to (ℝ, *): 1/x The identity element for (ℝ, +) is 1: a * 1 = 1 * a = aFor real number x the inverse is 1/x, because x * (1/x) = 1