Search results for "68t99"
showing 6 items of 6 documents
About Graph Mappings
2019
Summary In this articles adjacency-preserving mappings from a graph to another are formalized in the Mizar system [7], [2]. The generality of the approach seems to be largely unpreceeded in the literature to the best of the author’s knowledge. However, the most important property defined in the article is that of two graphs being isomorphic, which has been extensively studied. Another graph decorator is introduced as well.
About Supergraphs. Part III
2019
Summary The previous articles [5] and [6] introduced formalizations of the step-by-step operations we use to construct finite graphs by hand. That implicitly showed that any finite graph can be constructed from the trivial edgeless graph K 1 by applying a finite sequence of these basic operations. In this article that claim is proven explicitly with Mizar[4].
Underlying Simple Graphs
2019
Summary In this article the notion of the underlying simple graph of a graph (as defined in [8]) is formalized in the Mizar system [5], along with some convenient variants. The property of a graph to be without decorators (as introduced in [7]) is formalized as well to serve as the base of graph enumerations in the future.
About Vertex Mappings
2019
Summary In [6] partial graph mappings were formalized in the Mizar system [3]. Such mappings map some vertices and edges of a graph to another while preserving adjacency. While this general approach is appropriate for the general form of (multidi)graphs as introduced in [7], a more specialized version for graphs without parallel edges seems convenient. As such, partial vertex mappings preserving adjacency between the mapped verticed are formalized here.
Natural Addition of Ordinals
2019
Summary In [3] the existence of the Cantor normal form of ordinals was proven in the Mizar system [6]. In this article its uniqueness is proven and then used to formalize the natural sum of ordinals.
Some Remarks about Product Spaces
2018
Summary This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4]. Let {Ti}i∈I be a family of topological spaces. The prebasis of the product space T = ∏ i∈I Ti is defined in [5] as the set of all π −1 i (V) with i ∈ I and V open in Ti . Here it is shown that the basis generated by this prebasis consists exactly of the sets ∏ i∈I Vi with Vi open in Ti and for all but finitely many i ∈ I holds Vi = Ti . Given I = {a} we have T ≅ Ta , given I = {a, b} with a≠ b we have T ≅ Ta ×Tb . Given another family of topological spaces {Si}i∈I such that Si ≅ Ti fo…