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

By Stankey Burris, H. P. Sankappanavar

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2) The open subsets of a topological space with the ordering ⊆ form a complete lattice. (3) Su(I) with the usual ordering ⊆ is a complete lattice. A complete lattice may, of course, have sublattices which are incomplete (for example, consider the reals as a sublattice of the extended reals). It is also possible for a sublattice of a complete lattice to be complete, but the sups and infs of the sublattice not to agree with those of the original lattice (for example look at the sublattice of the extended reals consisting of those numbers whose absolute value is less than one together with the numbers −2, +2).

Then the kernel of α, written ker(α), is defined by ker(α) = { a, b ∈ A2 : α(a) = α(b)}. 8. Let α : A → B be a homomorphism. Then ker(α) is a congruence on A. Proof. If ai , bi ∈ ker(α) for 1 ≤ i ≤ n and f is n-ary in F, then αf A (a1 , . . , an ) = f B (αa1 , . . , αan) = f B (αb1 , . . , αbn ) = αf A (b1 , . . , bn ); hence f A (a1 , . . , an ), f A (b1 , . . , bn ) ∈ ker(α). Clearly ker(α) is an equivalence relation, so it follows that ker(α) is actually a congruence on A. 2 When studying groups it is usual to refer to the kernel of a homomorphism as a normal subgroup, namely the inverse image of the identity element under the homomorphism.

An )θf A (b1 , . . , bn ) holds. The compatibility property is an obvious condition for introducing an algebraic structure on the set of equivalence classes A/θ, an algebraic structure which is inherited from the algebra A. For if a1 , . . , an are elements of A and f is an n-ary symbol in F, then the easiest choice of an equivalence class to be the value of f applied to a1 /θ, . . , an /θ would be simply f A (a1 , . . , an )/θ. This will indeed define a function on A/θ iff (CP) holds.

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