Varieties of Algebras

Abstract

A non-empty set A is called a universal algebra with a set of operations, Ω if for any ω ∈ Ω there exists a natural number n = n(ω) (it is possible that n = 0) such that ω is an n-ary (or n-local) operation on A, i.e. ω is a mapping from ${A^n} = \underbrace {A \times \ldots \times A}_n$ into A. For n = 0 the mapping ω simply fixes an element from A (it can be assumed that ω ∈ A). For instance, a multiplicative group G with unit e and inverse element a ⁻¹ for a ∈ G is a universal algebra with set of operations Ω = {ω ₀, ω ₁, ω ₂} such that ω ₀ = e, ω ₁(a) = a ⁻¹, ω ₂(a, b) = ab (a, b ∈ G). It is clear that not each universal algebra with such a set of operations is a group: as a matter of fact, nothing is said about the properties of the operations.