Subdirect products and propagating equations with an application to Moufang theorem

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.