Day定理在$n$为偶数时是精确的
Day's Theorem is sharp for $n$ even
AI总结：
该研究解决了Maltsev条件相对长度问题，肯定回答了Day五十多年前的经典问题，证明当n为偶数时Day关于构造同余模项数量的结果精确，并处理多种项类型，结果适用于局部有限簇。
摘要：
我们解决了关于Maltsev条件相对长度的一些问题，特别是，我们对A. Day在五十多年前提出的一个经典问题给出了肯定回答。具体而言，同余分配和同余模簇都通过存在有限但数量可变且适当的项来承认Maltsev刻画。A. Day证明了从见证同余分配性的Jónsson项$t_0, \dots, t_n$可以构造见证同余模性的项$u_0, \dots, u_{2n-1}$。我们证明当$n$为偶数时，Day关于此类项数量的结果是精确的。我们还处理了其他类型的项，如alvin、Gumm、Pixley、directed、specular、mixed和defective。所有结果在限制于局部有限簇时均成立。我们引入了一些同余分配簇族，并刻画了它们满足的许多同余恒等式。
英文摘要：
We solve some problems about relative lengths of Maltsev conditions, in particular, we give an affirmative answer to a classical problem raised by A. Day more than fifty years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1} $ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, Pixley, directed, specular, mixed and defective. All the results hold when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.
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