On residual finiteness of direct products of algebraic systems
On residual finiteness of direct products of algebraic systems
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关于代数系统直积的残差有限性
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
N. Ruškuc
中科院分区：
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作者：
R. Gray;N. Ruškuc
众所周知，如果两个代数结构 A 和 B 是残差有限的，那么它们的直积也是残差有限的。在这里我们讨论这个陈述的反面。如果 A 和 B 包含幂等，这当然是正确的，这涵盖了群、环等的情况。我们证明，相反的情况也适用于半群，即使它们不需要具有幂等。我们还展示了三个例子，表明相反的情况一般不成立。
It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.