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arXiv 1201.2986math.COmath.LO
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两个关于反质体的禁导小定理

Two Forbidden Induced Minor Theorems for Antimatroids

Christian Joseph Altomare

备注：
Aside from the abstract, this paper was written before the author learned of antimatroids. Antimatroid theorists will no doubt recognize some of the basic lemmas as familiar. However, of the two main theorems, the proof of the first main theorem and statement and proof of the second main theorem are new
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摘要：

反质体由Dilworth在格的背景下发现[4]，并由Edelman和Jamison在[5]中作为凸几何引入。当前论文的作者在数理逻辑的证明系统中独立地发现了（可能是无限的）反物质[1]。逻辑学家Carlson在[2]中隐含地使用了这种证明系统作为可能无限的反矩阵的观点。虽然反拟阵在某种意义上是拟阵的对偶，但已知的反拟阵禁子定理却少得多。文[6]、[7]、[8]和[9]中证明了这种形式的一些结果。本文证明了两个禁止诱导小定理，这些对象，我们认为是证明系统。我们的第一个主要定理给出了偏序作为证明系统的禁止诱导次要特征的新证明，在[8]中在有限情况下证明，并在[10]中陈述，我们称之为强aut下降证明系统。它本质上是说，撇开病理学不谈，存在某个唯一的最简单非偏序集。我们的第二个主要定理陈述了一个新的结果，即除了病理学之外，存在一个包含点$x$和$y$的唯一的最简单的证明系统，使得$x$在一个上下文中需要$y$，而$y$在另一个上下文中需要$x$。

英文摘要：

Antimatroids were discovered by Dilworth in the context of lattices [4] and introduced by Edelman and Jamison as convex geometries in[5]. The author of the current paper independently discovered (possibly infinite) antimatroids in the context of proof systems in mathematical logic [1]. Carlson, a logician, makes implicit use of this view of proof systems as possibly infinite antimatroids in [2]. Though antimatroids are in a sense dual to matroids, far fewer antimatroid forbidden minor theorems are known. Some results of this form are proved in [6], [7], [8], and [9]. This paper proves two forbidden induced minor theorems for these objects, which we think of as proof systems. Our first main theorem gives a new proof of the forbidden induced minor characterization of partial orders as proof systems, proved in [8] in the finite case and stated in [10] for what we call strong aut descendable proof systems. It essentially states that, pathologies aside, there is a certain unique simplest nonposet. Our second main theorem states the new result that, pathologies aside, there is a certain unique simplest proof system containing points $x$ and $y$ such that $x$ needs $y$ in one context, yet $y$ needs $x$ in another.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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