By Stankey Burris, H. P. Sankappanavar

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Extra resources for A Course in Universal Algebra

Example text

Let P be a poset such that A exists for every subset A, or such that exists for every subset A. Then P is a complete lattice. A Proof. Suppose A exists for every A ⊆ P. Then letting Au be the set of upper bounds of A in P, it is routine to verify that Au is indeed A. 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.

Let α : A → B be an embedding. Then for an n-ary function symbol f and a1 , . . , an ∈ A, f B (αa1 , . . , αan ) = αf A (a1 , . . , an ) ∈ α(A), hence α(A) is a subuniverse of B. 5. If α : A → B is an embedding, α(A) denotes the subalgebra of B with universe α(A). A problem of general interest to algebraists may be formulated as follows. Let K be a class of algebras and let K1 be a proper subclass of K. ) Two basic questions arise in the quest for structure theorems. (1) Is every member of K isomorphic to some member of K1 ?

C is a closure operator, and from the definition of compact elements it follows that C is algebraic. The map a → {b ∈ A : b ≤ a} gives the desired isomorphism as L is compactly generated. 2 References 1. M. Cohn [9] 2. A. Tarski [1930] Exercises §5 1. If G is a group and X ⊆ G, let C(X) be the subgroup of G generated by X. Show that C is an algebraic closure operator on G. 2. If G is a group and X ⊆ G, let C(X) be the normal subgroup generated by X. Show that C is an algebraic closure operator on G.

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

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