关于代数嵌入的一些结果，在de Bruijn和McKenzie之后
Some results on embeddings of algebras, after de Bruijn and McKenzie
摘要：
1957年，N. G. de Bruijn证明了无限集Ω上的对称群Sym（Ω）包含一个2^{card（Ω）}生成元上的自由子群，并证明了一个更一般的陈述，其一个示例结果是，对于基数为\leq card（Ω）的任何群A，Sym（Ω）包含A的2^{card（Ω）}个副本的余积，不仅在所有群的簇中，而且在A所属的任何群的簇中。他的关键引理在这里推广到任意代数簇\bf{V}，并制定为一个关于函子集--> \bf{V}的声明。由此，我们可以很容易地得到上述结果的类似物，其中“群”和Sym（Ω）被“幺半群”和Ω的内映射的幺半群Self（Ω），被“结合K-代数”和具有基Ω的K-向量空间V的自同态的K-代数End_K（V），以及被“格”和Ω上的等价关系的格Equiv（Ω）所代替。推广了de Bruijn在1957年的论文中的另一个结果，证明了Sym（Ω），Self（Ω）和End_K（V）中的每一个都包含2^{card（Ω）}个自身拷贝的余积.该论文还给出了基数为2^{card（Ω）}的群{ \em not}可嵌入Sym（Ω）的例子，并且R.麦肯齐随后建立了一个大类这样的例子。这些结果被证明是Sym（Ω）中具有常数的方程组的Sym（Ω）中的解集格的一般性质的实例。同样，对于Self（Ω）、End_K（V）和Equiv（Ω），以及Ω上二元关系的幺半群\Rel，也得到了类似的结果--这一次强度不同。许多开放的问题和进一步调查的领域指出。
英文摘要：
In 1957, N. G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ωcontains a free subgroup on 2^{card(Ω)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(Ω), Sym(Ω) contains a coproduct of 2^{card(Ω)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(Ω) replaced by "monoid" and the monoid Self(Ω) of endomaps of Ω, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis Ω, and by "lattice" and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K (V) contains a coproduct of 2^{card(Ω)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(Ω)} that was {\em not} embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results -- this time of varying strengths -- are obtained for Self(Ω), End_K (V), and Equiv(Ω), and also for the monoid \Rel of binary relations on Ω. Many open questions and areas for further investigation are noted.
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