Algebra I Math 200 Robert Boltje

1 Semigroups and Monoids
1.1 Definition Let S be a set.
(a) A binary operation on S is a map b : S × S → S. Usually, b(x, y) is
abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc.
(b) Let (x, y) 7→ x ∗ y be a binary operation on S.
(i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z ∈ S.
(ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x, y ∈ S.
(iii) An element e ∈ S is called a left (resp. right) identity, if e ∗ x = x
(resp. x ∗ e = x) for all x ∈ S. It is called an identity element if it is a left
and right identity.
(c) The set S together with a binary operation ∗ is called a semigroup if
∗ is associative. A semigroup (S, ∗) is called a monoid if it has an identity
element.
1.2 Examples (a) Addition (resp. multiplication) on N0 = {0, 1, 2, . . .} is
a binary operation which is associative and commutative. The element 0
(resp. 1) is an identity element. Hence (N0, +) and (N0, ·) are commutative
monoids. N := {1, 2, . . .} together with addition is a commutative semigroup,
but not a monoid. (N, ·) is a commutative monoid.
(b) Let X be a set and denote by P(X) the set of its subsets (its power set).
Then, (P(X), ∪) and (P(X), ∩) are commutative monoids with respective
identities ∅ and X.
(c) x∗y := (x+y)/2 defines a binary operation on Q wh