Approximate satisfaction of identities
by
Walter Taylor
University of Colorado

For a topological algebra A based on a metric space A = (A, d), and for S a set of equations in the appropriate language, we define l_A(S) to be the sup of d(s^A(a), t^A(a)), over all s ≈ t ∈ S and over all appropriate a. Then, for a metric space A, l_A(S) is defined as the inf of l_A(S), over all topological algebras on A. l_A(S) measures how far the equations S must deviate from being satisfied on A with continuous operations. It is zero if S is so satisfiable, but may also be zero in the contrary case, even for compact A. This is an ongoing work on elementary calculations of, and facts about, l.

We know l_A completely for A an n-sphere (except for n = 3 and n=7): l_A(S) is either 0 or the diameter of Sⁿ. The rest of our knowledge is sporadic. l_A is not a topological invariant of A, but depends heavily on the metric of A. Except that, for A compact, the condition l_A(S) ≠ 0 is a topological invariant (in which case there may or may not be a uniform bound of l_A(S)/diam(A) away from zero).

For certain A, such as a closed unit interval, we have: if l_A(G×D) < e, then l_A(G) < 4e or l_A(D) < 4e. (Here × denotes the varietal product (the meet in the interpretability lattice).) This is an analog, for approximate satisfaction, to the known result, for topological algebras, that if [0, 1] models G×D, then [0, 1] models G or [0, 1] models D.

Let A be the realization of a finite simplicial complex, and let a be a computable real number. By simplicial approximation, the set of all finite S (in a fixed alphabet) such that l_A(S) < a is recursively enumerable.

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Date received: May 30, 2008