The Initial Algebra Theorem by Trnkov\'a et al.~states, under mild assumptions, that an endofunctor has an initial algebra provided it has a pre-fixed point. The proof crucially depends on transfinitely iterating the functor and in fact shows that, equivalently, the (transfinite) initial-algebra chain stops. We give a constructive proof of the Initial Algebra Theorem that avoids transfinite iteration of the functor. For a given pre-fixed point $A$ of the functor, it uses Pataraia's theorem to obtain the least fixed point of a monotone function on the partial order formed by all subobjects of $A$. Thanks to properties of recursive coalgebras, this least fixed point yields an initial algebra. We obtain new results on fixed points and initial algebras in categories enriched over directed-complete partial orders, again without iteration. Using transfinite iteration we equivalently obtain convergence of the initial-algebra chain as an equivalent condition, overall yielding a streamlined version of the original proof.
翻译:Trnkov\'a' et al. ~ states 的初始代数理论由 Trnkov\' a et al. ~ states 根据温和假设, 端点有一个初始代数, 只要它有一个前固定点, 端点就有一个初始代数。 证据关键地取决于对配方的瞬间迭代, 并且事实上表明, 等量的( 纯度) 初始代数链站点。 我们提供了避免对配方进行跨倍迭代的初始代数初始代数的建设性证据。 对于给定的给定前固定点( $A$ ), 它使用 Pataraia 的代数在由所有子项( $) 构成的部分顺序上获得最小的单质函数固定点 。 由于循环性煤值的特性, 这个最小的端点产生初始代数。 我们获得了固定点和初始代数的代数的分类的新结果, 并且不重复。 对于给定点和纯部分订单的类别, 它使用一个我们相当的半定点来获得原始精度验证的原始精度精度精度精度精度精度精度精度精度精准精准的精度, 。