# Download PDF by Alfred Bray Kempe: How to Draw a Straight Line: A Lecture on Linkages

By Alfred Bray Kempe

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1 LINDENBAUM ALGEBRAS The constructions we have just given play an entirely analogous role with respect to Heyting algebras to that of Lindenbaum algebras with respect Categorical Preliminaries 43 to Boolean algebras. In fact, The Lindenbaum algebra BT in the variables {Ai} with respect to the axioms T is just our HT∪T1, where T1 is the set of all formulas of the form ¬¬F→F, since the additional axioms of T1 are the only ones that need to be added in order to make all classical tautologies provable.

In algebraic topology, cartesian closed categories are particularly easy to work with. Neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed. Substitute categories have therefore been considered: the category of compactly generated Hausdorff spaces is cartesian closed, as is the category of Frölicher spaces. , the objects are the cpos, and the morphisms are the Scott continuous maps). [3] A Heyting algebra is a Cartesian closed (bounded) lattice.

Since for any a and b in a Heyting algebra H we have a ≤ b if and only if a b=1, it follows from 1 2 that whenever a formula F → G is provably true, we have F(a1, a2, …, an) =< G(a1, a2, …, an) for any Heyting algebra H, and any elements a1, a2, …, an ∈ H. (It follows from the deduction theorem that F G is provable if and only if G is provable from F, that is, if G is a provable consequence of F. In particular, if F and G are Categorical Preliminaries 39 provably equivalent, then F(a1, a2, …, an) ≤ G(a1, a2, …, an), since ≤ is an order relation.