Math 323 Properties of Integers 2015 Background: Definition of semigroup. Definition of group. A semigroup is a nonempty set S with a binary operation * which A group is a nonempty set S with a binary operation * which has has the following properties: 1 and 2 the following properties 1, 2, 3, and 4: 1. For each a, b in S, a*b is in S. 1. For each a, b in S, a*b is in S. 2. The operation is associative: 2. The operation is associative: For each a, b, c in S, (a*b)*c = a*(b*c). For each a, b, c in S, (a*b)*c = a*(b*c). [Thus, a group starts its life as a semigroup, but a group has (We denote the binary operation here by *, but can have many other additional properties given below.] notations, such as + or . or o or × , etc., or no symbol at all (i.e., 3. There is an identity element in S, which we will call e here, such simply by “juxtaposition”) as we usually do with ordinary that for each a in S, a e = e a = a. multiplication.), * * In a semigroup S, an element e in S is called an identity, or an 4. For each a in S, there is an inverse element a-1 in S such that identity element, provided that a*a-1 = a-1*a = e . for each a in S, a*e = e*a = a. Most mathematicians don't REQUIRE a semigroup to have an (The identity element is often denoted by a different symbol such as identity element, but, hey, if you’ve got it, use it. 0 or 1 or I, etc.) -1 Theorem: If a semigroup has an identity element, then it has only The inverse element a is sometimes denoted by -a. one. (Proved in class and in various contexts in the Class Notes.) For example, when the operation is +, the identity is denoted by 0 and the inverse is denoted by -a, as we all know.

If the operation * is commutative, so a*b = b*a for all a and b If the operation * is commutative, then the group is called a in the semigroup, then the semigroup is called, believe it or not, a commutative group. commutative semigroup. ------Properties of the set of integers, Z, with the operations of addition, +, and multiplication, . Z with the operation + is a commutative group, with identity 0. Z with the operation . is a commutative semigroup with identity 1. Interaction of addition and multiplication: Distributive property: For all integers a, b, and c, a.(b + c) = ab + ac. No divisors of zero property: For all integers a and b, if a.b = 0, then a = 0 or b = 0.