ME(EE) 550 Foundations of Engineering Systems Analysis Chapter Two: Algebraic Structure of Vector Spaces

Definition 1.2. (Semigroup) A binary algebra (S;⊛) is called a semigroup if it satisfies the associativity property, i.e., (α ⊛ β)⊛ γ = α⊛ (β ⊛ γ) ∀α, β, γ ∈ S. A semigroup is called commutative if α⊛ β = β ⊛ α ∀α, β ∈ S. Definition 1.3. (Identity Element) An element 1l ∈ S is said to be a left identity of the binary algebra (S;⊛) if 1l ⊛ α = α ∀α ∈ S. Similarly, an element 1r ∈ S is said to be a right identity of the binary algebra (S;⊛) if α ⊛ 1r = α ∀α ∈ S. Finally, an element 1 ∈ S is said to be an identity of the binary algebra (S;⊛) if 1⊛ α = α⊛ 1 = α ∀α ∈ S. Definition 1.4. (Zero Element) An element 0l ∈ S is said to be a left zero of the binary algebra (S;⊛) if 0l ⊛ α = 0l ∀α ∈ S. Similarly, an element 0r ∈ S is said to be a right zero of the binary algebra (S;⊛) if α ⊛ 0r = 0r ∀α ∈ S. Finally, an element 0 ∈ S is said to be a zero of the binary algebra (S;⊛) if 0⊛ α = α⊛ 0 = 0 ∀α ∈ S.