A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a “quasi-coincidence” point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of “quasi-coincidence” points is bounded and has a lower bound, which also is a “quasi-coincidence” point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions.

中文翻译：

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从局部有序空间到任意集的两个映射的重合点

一对映射的重合点是一个元素，这些映射在这些元素处具有相同的值。AV Arutyunov，ES Zhukovskiy和SE Zhukovskiy研究了部分有序空间的映射的重合点（请参见《拓扑结构及其应用》，2015年，第179卷，第1期，第13-33页）；他们特别证明，从部分有序空间到部分有序空间的有序覆盖映射和单调映射都具有重合点。在本文中，我们研究了从局部有序空间到集合的一对映射的重合点的存在，其中没有定义二元关系，因此，不可能定义映射的覆盖度和单调性。为了研究上述问题，我们定义了“准巧合”点的概念。我们将其理解为一个元素，为此存在另一个元素，该元素不超过初始元素，并且该元素的第一个映射的值等于初始元素处的第二个映射的值。事实证明，以下条件足以满足巧合点的存在：任何“准巧合”点链都是有界的且具有下限，这也是“准巧合”点。我们给出了一个满足上述要求的映射示例，并且不允许应用针对有序覆盖和单调映射的重合点获得的结果。另外，我们给出了映射的重合点的稳定性概念的解释，这些重合点作用于部分有序空间中的小扰动，并建立了相应的稳定性条件。