Some Connections between Residual Finiteness, Finite Embeddability and the Word Problem
Some Connections between Residual Finiteness, Finite Embeddability and the Word Problem
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残差有限性、有限可嵌入性和词问题之间的一些联系
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发表时间：
1969
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通讯作者：
T. Evans
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作者：
T. Evans
有限可嵌入性。一个代数A是剩余有限的，如果对A中的任意x#y，有A到有限代数的同态a，使得XCT#Yu。关于簇中的不完全或部分代数的概念，我们参考文献[4，6]。我们称群V中的代数A具有有限可嵌入性，如果A中包含的任何有限不完全F-代数可嵌入到有限F-代数中。称簇V具有有限可嵌入性，如果V中的每个代数都具有有限可嵌入性。因此，如果任何可嵌入的有限不完全K-代数可嵌入到有限F-代数中，则簇V具有有限可嵌入性。我们还注意到，如果簇的有限生成代数具有有限可嵌入性，则簇具有有限可嵌入性。为此，设A是V中的代数，其有限生成的代数具有有限可嵌入性，I是A中包含的有限不完全代数，B是由F-代数自由生成的。代数B不一定是A的子代数，但B包含/同构[4；引理1]，且B是有限生成的。因此，I是有限可嵌入的，因此V具有有限可嵌入性。
Finite embeddability. An algebra A is residually finite if for any x # y in A, there is a homomorphism a of A onto a finite algebra such that xct # yu. For the notion of an incomplete or partial algebra in a variety, we refer to [4, 6]. We say that an algebra A in a variety V has the finite embeddability property if any finite incomplete F-algebra contained in A is embeddable in a finite F-algebra. A variety V is said to have the finite embeddability property if every algebra in V has the property. Thus, a variety V has the finite embeddability property if any finite incomplete K-algebra which is embeddable is embeddable in a finite F-algebra. We note also that a variety has the finite embeddability property if its finitely generated algebras have this property. To see this, let A be an algebra in a variety V whose finitely generated algebras have the finite embeddability property and let / be a finite incomplete algebra contained in A. Let B be the F-algebra freely generated by / . The algebra B is not necessarily a subalgebra of A but B contains / isomorphically [4; Lemma 1] and B is finitely generated. Hence, I is finitely embeddable and so V has the finite embeddability property.