A semi-group is a non-empty set G with an associative binary operation.

Example: Set of natural numbers ℕ = {1, 2, 3, ...} under addition; Addition is associative: (1+2)+3=1+(2+3); so, (ℕ,+) is a semigroup.

A monoid is a semigroup with identity.

Example: Set: ℕ₀ = {0, 1, 2, 3, ...} under addition. Associativity: (0+1)+2=0+(1+2); Identity Element: 0, because for any a in ℕ₀, a + 0 = 0 + a = a (ex. 1+0=1) ; so, (ℕ₀,+) is a monoid.

A group is a monoid with inverse element . Satisfies the four properties ® C.A.I.N.

Example: Set of all integers. Set: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...} Operation: Addition (+)

A group in which the operation is commutative. Satisfies all the group properties (C.A.I.N) and the commutativity.

Example: (Z,+) — the set of integers under addition.

A group that has a finite number of elements.

Example: The set {0,1,2} with addition modulo 3 forms a finite group.

A group where all the elements can be generated by repeatedly applying the group operation to a single element.