论文标题
保持自由产品的自相似性
Preserving self-similarity in free products of semigroups
论文作者
论文摘要
在自动机半群的免费产品下，我们改善了闭合的早期结果。我们考虑部分自动机，并表明两个自相似的半群（或自动机半群）的自由产物是自相似的（一个自动机半群），如果从一个基础半群到另一个基础半群中存在同态。所使用的构造是可计算的，并产生进一步的后果。其中之一是，我们可以将自由发电机与任何相似的半群（或自动机半群）相结合，并保留自相似性的属性（或自动机半群）。 两个半群之间的同态存在是非常宽松的要求。特别是，如果其中一个半群包含一个愿意，则可以满足满足。为了探索此要求的限制，我们表明没有简单或$ 0 $ -SIMPLE的无掌式半群是有限生成的自相似的半群（或自动机半群）。此外，我们举例说明了一对残留有限的半群，而没有同构的同构。
We improve on earlier results on the closure under free products of the class of automaton semigroups. We consider partial automata and show that the free product of two self-similar semigroups (or automaton semigroups) is self-similar (an automaton semigroup) if there is a homomorphism from one of the base semigroups to the other. The construction used is computable and yields further consequences. One of them is that we can adjoin a free generator to any self-similar semigroup (or automaton semigroup) and preserve the property of self-similarity (or being an automaton semigroup). The existence of a homomorphism between two semigroups is a very lax requirement; in particular, it is satisfied if one of the semigroups contains an idempotent. To explore the limits of this requirement, we show that no simple or $0$-simple idempotent-free semigroup is a finitely generated self-similar semigroup (or an automaton semigroup). Furthermore, we give an example of a pair of residually finite semigroups without a homomorphism from one to the other.
