The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.

翻译：著名的Kleene不动点定理在指称语义学中递归规范的数学建模中至关重要。本文探讨了上述结果的假设是否可以弱化。我们对该问题给出了肯定回答，从而刻画了自映射为保证其不动点集非空所需满足的性质，且无需假设偏序集满足任何完备性概念。此外，深入研究了偏序集来自拟度量空间的情形。最后，得到了所提出理论的一个应用：具体而言，提出了一种数学方法，用于讨论运行时间满足递推方程的算法的渐近复杂度。该方法可以复现文献中用于算法渐近复杂度分析的不动点基方法。然而，我们的新方法改进了上述方法，因为它比文献中假设的条件要求更少，并且允许同时给出算法运行时间的上界和下界渐近界。