Algebraic Structures

In this text, we focus on operations of arity 2, 1, and 0. – For n = 2, f : A → A is a binary operation and is usually written in infix notation, using a binary operation symbol like ·, ∗, or +. Hence, instead of f(a1, a2) we write a1fa2. – For n = 1, f : A→ A is a unary operation. – For n = 0, f : A → A is a nullary operation or a constant. An algebra (or an algebraic structure) is a set A, the carrier, together with a set of operations onA. In addition, the operations may be required to satisfy a set of equations (identities). Let us take a closer look at nullary operations and clarify the term “constants”. As a matter of convention, A is a singleton set, usually denoted as {∗}. Hence, a nullary operation is a function f : {∗} → A and it is uniquely determined by the image f(∗) which is a distinguished element of A, i.e., a constant.