论文标题
分区产品和传播方程，并应用于Moufang定理
Subdirect products and propagating equations with an application to Moufang theorem
论文作者
论文摘要
我们介绍了传播方程式的概念，并专注于在各种循环中传播关联的情况。 如果每当$ \ varepsilon（\ varepsilon $ \ varepsilon（\ costrightArrow x）$持有，并且$ \ overrightArrow y $均包含$ x $ x $ x $ x $ x $ x $ x $ x $ x，则方程$ \ varepsilon $在代数$ x $中传播。如果$ \ varepsilon $ in $ x $繁殖，则它将在所有亚代词和$ x $的产品中繁殖，但不一定在所有同构图像中的$ x $。如果$ \ mathcal v $是一种多样性，那么传播核心$ \ Mathcal V _ {[\ Varepsilon]} = \ {x \ in \ Mathcal V：\ varepsilon $ in $ x \ \} $ specagates $ specagaTes $是一种难题，但不一定是一种品种。 我们通过基础证明了Goursat的Lemma用于循环的引理，并描述了$ X^K $的所有细分产品，以及所有有限生成的LOOPS in $ \ MATHBF {hsp}（X）$用于非ABELIAN简单Loop $ x $。如果$ \ MATHCAL V $是各种循环，可以在其中传播关联性，则$ x $是一个有限的环路，在该环路中，关联性繁殖和每一个$ x $的每个subloop都是简单的，或包含在$ \ MATHCAL V $中，然后是$ \ \ alsbf {hsp}（hsp}（hsp}（x）（x）\ lor \ lor \ lor \ lor \ lor \ lor \ v $ \ lor \ v $ \ lor \ v $ \ lor \ v $ \ lor v $ 我们研究了Steiner Loops of Associativity的传播核心$ \ Mathcal S _ {[x（yz）=（xy）z]} $。虽然这不是一个品种，但我们展示了$ \ Mathcal s _ {[x（yz）=（xy）z]} $中包含的许多品种，每个品种都为Rajah的问题提供了解决方案，即在Moufang loops中不包含的各种环。
We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $\varepsilon$ propagates in an algebra $X$ if $\varepsilon(\overrightarrow y)$ holds whenever $\varepsilon(\overrightarrow x)$ holds and the elements of $\overrightarrow y$ are contained in the subalgebra of $X$ generated by $\overrightarrow x$. If $\varepsilon$ propagates in $X$ then it propagates in all subalgebras and products of $X$ but not necessarily in all homomorphic images of $X$. If $\mathcal V$ is a variety, the propagating core $\mathcal V_{[\varepsilon]} = \{X\in\mathcal V:\varepsilon$ propagates in $X\}$ is a quasivariety but not necessarily a variety. We prove by elementary means Goursat's Lemma for loops and describe all subdirect products of $X^k$ and all finitely generated loops in $\mathbf{HSP}(X)$ for a nonabelian simple loop $X$. If $\mathcal V$ is a variety of loops in which associativity propagates, $X$ is a finite loop in which associativity propagates and every subloop of $X$ is either nonabelian simple or contained in $\mathcal V$, then associativity propagates in $\mathbf{HSP}(X)\lor\mathcal V$. We study the propagating core $\mathcal S_{[x(yz)=(xy)z]}$ of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in $\mathcal S_{[x(yz)=(xy)z]}$, each providing a solution to Rajah's problem, i.e., a variety of loops not contained in Moufang loops in which Moufang Theorem holds.
