Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interc
Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interc
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关系代数、有限维圆柱代数及其相互关系入门课程
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发表时间：
1990
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通讯作者：
R. Maddux
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作者：
R. Maddux
这些是在代数关系代数会议上发表的关于关系代数、nite维柱面代数及其相互关系的短期课程的笔记。关系代数（RA）与柱面代数（CA）和多进等式代数（PEA）密切相关，但在某些方面它们是非常不同的。这些联系和对话使这些主题之间能够进行建设性的互动。通过考虑二元关系族和二元关系上的某些运算，关系代数自然产生。这些操作中最关键的是相对乘法（最初被德·摩根称为复合，D1856，D1864），它需要在定义中引用第三个对象。相对乘法是一种结合运算。这一事实的证明需要对四个对象进行检查。这些非正式的观察反映在以下数学事实中：关系代数产生于那些由其二维元素生成的圆柱代数和多进代数，它们至少有三维（因此相对乘法是deenable），并且满足所有可用四维证明的恒等式（因此相对乘法是结合的）。相对乘法的可判定性是区分3维以上的圆柱代数和多进代数与低维代数的一个重要特征。事实上，CA和PEA只是在情况3中有不可判定的方程理论。此外，如果3，那么在这些变种中的每一个中，可表示代数都形成一个不可完全公理化的真子变种，但只要2就不是这样。类似地，RA具有不可判定的方程理论，并且可表示的关系代数形成RA的非无限基真子簇。因此，在这些注释的其余部分中，我们假设3。
These are notes for a short course on relation algebras, nite-dimensional cylindric algebras, and their interconnections, delivered at the Conference on Alge-Relation algebras (RA's) are closely linked to cylindric algebras (CA's) and polyadic equality algebras (PEA's), and yet in certain ways they are quite diierent. The links and diierences allow for constructive interaction between these subjects. Relation algebras arise naturally by considering families of binary relations and certain operations on binary relations. The most crucial of these operations, relative multiplication (originally called composition by De Morgan D1856], D1864]), requires a reference to a third object in its deenition. Relative multiplication is an associative operation. The proof of this fact requires the examination of four objects. These informal observations are reeected in the following mathematical facts: relation algebras arise from those cylindric and polyadic algebras which are generated by their two-dimensional elements, which have at least three dimensions (so that relative multiplication is deenable), and which satisfy all the identities provable with four dimensions (so that relative multiplication turns out to be associative). The deenability of relative multiplication is a crucial feature which distinguishes cylindric and polyadic algebras of dimension 3 or more from those of lower dimension. Indeed, CA and PEA have undecidable equational theories just in case 3. Furthermore, if 3 then in each of these varieties the representable algebras form a proper subvariety which is not nitely axiomatizable, but this is not so whenever 2. Similarly, RA has an undecidable equational theory and the representable relation algebras form a nonnnitely based proper subvariety of RA. Therefore, throughout the rest of these notes we assume 3.