By Stankey Burris, H. P. Sankappanavar

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A group G is Abelian (or commutative ) if the following identity is true: §1. Definition and Examples of Algebras 27 G4: x · y ≈ y · x. Groups were one of the earliest concepts studied in algebra (groups of substitutions appeared about two hundred years ago). The definition given above is not the one which appears in standard texts on groups, for they use only one binary operation and axioms involving existential quantifiers. The reason for the above choice, and for the descriptions given below, will become clear in §2.

Then Sg is an n-ary closure operator on A. Proof. 2) E(X) ⊆ (Sg)n (X) ⊆ Sg(X); hence Sg(X) = X ∪ E(X) ∪ E 2 (X) ∪ · · · ⊆ (Sg)n (X) ∪ (Sg)2n (X) ∪ · · · ⊆ Sg(X), so 36 II The Elements of Universal Algebra Sg(X) = (Sg)n (X) ∪ (Sg)2n (X) ∪ · · · . 3. Suppose C is a closure operator on S. A minimal generating set of S is called an irredundant basis. Let IrB(C) = {n < ω : S has an irredundant basis of n elements}. The next result shows that the length of the finite gaps in IrB(C) is bounded by n − 2 if C is an n-ary closure operator.

In view of (a) this leads to Θ(a1 , . . , an) ⊆ Θ(a1 , a2 ) ∨ · · · ∨ Θ(an−1 , an ), so Θ(a1 , . . , an ) = Θ(a1 , a2 ) ∨ · · · ∨ Θ(an−1 , an ). (d) For a, b ∈ θ clearly a, b ∈ Θ(a, b) ⊆ θ so θ⊆ {Θ(a, b) : a, b ∈ θ} ⊆ {Θ(a, b) : a, b ∈ θ} ⊆ θ; hence θ= {Θ(a, b) : a, b ∈ θ} = {Θ(a, b) : a, b ∈ θ}. 5 because in 1963 Gr¨atzer and Schmidt proved that for every algebraic lattice L there is an algebra A such that L ∼ = Con A. Of course, for particular classes of algebras one might find that some additional properties hold for the corresponding classes of congruence lattices.

### A Course in Universal Algebra by Stankey Burris, H. P. Sankappanavar

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