By Stankey Burris, H. P. Sankappanavar

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The other half of the theorem is proved similarly. 2 In the above theorem the existence of ∅ guarantees a largest element in P, and likewise the existence of ∅ guarantees a smallest element in P. 2 would be to say that P is complete if it has a largest element and the inf of every nonempty subset exists, or if it has a smallest element and the sup of every nonempty subset exists. Examples. (1) The set of extended reals with the usual ordering is a complete lattice. (2) The open subsets of a topological space with the ordering ⊆ form a complete lattice.

An ). (∗) We say A is isomorphic to B, written A ∼ = B, if there is an isomorphism from A to B. If α is an isomorphism from A to B we may simply say “α : A → B is an isomorphism”. As is well-known, following Felix Klein’s Erlanger Programm, algebra is often considered as the study of those properties of algebras which are invariant under isomorphism, and such properties are called algebraic properties. Thus from an algebraic point of view, isomorphic algebras can be regarded as equal or the same, as they would have the same algebraic structure, and would differ only in the nature of the elements; the phrase “they are equal up to isomorphism” is often used.

If L is a lattice and A ⊆ L, let u(A) = {b ∈ L : a ≤ b for a ∈ A}, the set of upper bounds of A, and let l(A) = {b ∈ L : b ≤ a for a ∈ A}, the set of lower bounds of A. Show that C(A) = l(u(a)) is a closure operator on A, and that the map α : a → C({a}) gives an embedding of L into the complete lattice LC (called the Dedekind-MacNeille completion). What is the Dedekind-MacNeille completion of the rational numbers? 5. If we are given a set A, a family K of subsets of A is called a closed set system for A if there is a closure operator on A such that the closed subsets of A are precisely the members of K.

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

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