Universal Algebra

Abstract

An n -ary operation (n ∈ ℕ₀) on a set A is a map ω: A ⁿ → A, where A ⁰ := { ∅ }. The number n is called the arity of ω. A universal algebra is a pair 𝒜 = (A, Ω) consisting of a non-empty set A and a set Ω of operations on A. The set A and the members of Ω are called the universe and the fundamental operations of the algebra 𝒜, respectively. In practice, one usually is not interested in a single, isolated algebra but in a class of algebras of the same type. Therefore it is more customary to consider the set Ω not as the set of operations on the given set A but rather as the set of operation symbols. Formally this is achieved by first introducing the notion of type. The type is a set Ω together with a partition Ω = Ω⁰ ∪ Ω¹ ∪ Ω²⋯. (Empty Ωⁱ are allowed.) Alternatively, the type is a set Ω together with a mapping r: Ω → ℕ⁰. A universal algebra of type Ω, or simply an Ω-algebra is a pair 𝒜 = (A, Ω) where A is a non-empty set and to every ω ∈ Ωⁿ it is assigned an n-ary operation on A, denoted by the same symbol ω. Cf. Section G.10 for a generalization of these concepts.