一元代数及其直积中的剩余紧性及相关性质
Residual Finiteness and Related Properties in Monounary Algebras and their Direct Products
摘要：
本文从剩余有限性、强/弱子代数可分性和完全可分性等性质出发，讨论了一元代数的直积与其分支之间的关系。对于每一个性质$\mathcal{P}$，我们给出了一个图判据$\mathcal{C_P}$，使得一元代数$A$具有性质$\mathcal{P}$当且仅当它满足$\mathcal{C_P}$ .我们还证明了对于一元代数的直积$A\times B$，$A\times B$具有$\mathcal{P}$性质当且仅当下列条件之一为真：或者$A$和$B$都具有$\mathcal{P}$性质，或者$A$和$B$中至少有一个是后向有界的，这是一种特殊的性质，它支配着直积，并保证所有的$\mathcal{P}$成立.
英文摘要：
In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties $\mathcal{P}$, we give a graphical criterion $\mathcal{C_P}$ such that a monounary algebra $A$ has property $\mathcal{P}$ if and only if it satisfies $\mathcal{C_P}$. We also show that for a direct product $A\times B$ of monounary algebras, $A\times B$ has property $\mathcal{P}$ if and only if one of the following is true: either both $A$ and $B$ have property $\mathcal{P}$, or at least one of $A$ or $B$ are backwards-bounded, a special property which dominates direct products and which guarantees all $\mathcal{P}$ hold.
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