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

By Stankey Burris, H. P. Sankappanavar

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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.

Then a function α : A → B is an isomorphism from A to B if α is one-to-one and onto, and for every n-ary f ∈ F, for a1 , . . , an ∈ A, we have αf A (a1 , . . , an ) = f B (αa1 , . . , αan ). (∗) 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.

An algebra A, ∨, ∧, , 0, 1 with two binary, one unary, and two nullary operations is an n-valued Post algebra if it satisfies every identity satisfied by the algebra Pn = {0, 1, . . , n − 1}, ∨, ∧, , 0, 1 where {0, 1, . . , n − 1}, ∨, ∧, 0, 1 is a bounded chain with 0 < n − 1 < n − 2 < · · · < 2 < 1, and 1 = 2, 2 = 3, . . , (n − 2) = n − 1, (n − 1) = 0, and 0 = 1. See Figure 8, where the unary operation is depicted by arrows. In IV§7 we will give a structure theorem for all n-valued Post algebras, and in V§4 show that they can be defined by a finite set of equations.

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