由恒等式定义的数学结构
Mathematical Structures Defined by Identities
摘要：
我们提出的论点，有一个限制的数量可能的结构，公理赋予涉及操作的身份。在代数的情况下，一个二元运算满足一个形式上可约（待解释）单位之间的两个n-迭代的操作，它是建立这样的代数的频率趋于零，因为n趋于无穷大。这是证明了一个适当的顺序和标签的表达式（字）的相应的自由代数和形成一系列的tableaux的条目是标签。表格揭示了令人惊讶的对称性，用加泰罗尼亚数及其分区表示。由于定义的身份和tableaux所有迭代的顺序高于n落入等价类的语义相等的。依赖于表格的类数计算为所有代数的顺序n = 3（和部分n = 4）。某些类数是不变量，因为它们对于同阶代数是相等的。代数与任何arity的几个操作被认为是。提出了依赖于结构同态的Catalan数的一个推广，并建立了相应的生成函数。作为这方面的一个例子，一种绞多项式构造表征正式建立的迭代。由于没有区别的各种代数，同构或不，这是模型的身份的结果也可以制定方面的品种与签名的身份问题。
英文摘要：
We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to be explained) identity between two n-iterates of the operation, it is established that the frequency of such algebras goes to zero as n tends to infinity. This is proved by a suitable ordering and labeling of the expressions (words) of the corresponding free algebra and the formation of a series of tableaux whose entries are the labels. The tableaux reveal surprising symmetry properties, stated in terms of the Catalan numbers and their partitions. As a result of the defining identity and the tableaux all iterates of order higher than n fall into equivalence classes of semantically equal ones. Classnumbers depending on the tableaux are calculated for all algebras of order n = 3 (and partially for n = 4). Certain classnumbers are invariants in the sense that for algebras of same order they are equal. Algebras with several operations of any arity are considered. A generalization of Catalan numbers depending on homorphisms of the structure is proposed and corresponding generating functions set up. As an example of this, a kind of skein polynomials are constructed characterizing the formal build-up of iterates. As no distinction is made between the various algebras, isomorphic or not, which are models of the identity the results can also be formulated in terms of varieties with signature the identity in question.
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