关于Tarski意义下的素半群以及Tarski和Rhodes意义下的素群
On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes
AI总结：
本文探讨了在Tarski意义下素半群的存在性，证明非空半群类别中不存在素对象，但对单体和某些半群子类别给出了素对象的例子，并讨论了素性条件在群类别中的关系。
摘要：
如果$\mathcal{C}$是一个闭合于有限直积的代数范畴，且$M_\mathcal{C}$是$\mathcal{C}$中成员的同构类的交换半群，其运算由直积诱导，则A.Tarski定义了非恒等元素$p$为素元，当且仅当在该半群中，若$p$整除两个元素的乘积，则必整除其中一个元素。并称$\mathcal{C}$中的对象为素对象，若其同构类具有此性质。McKenzie, McNulty和Taylor询问非空半群类别是否存在素对象。本文第二部分证明其不存在。然而，在单体类别和某些半群子类别中，第三至第四部分给出了素对象的例子。第五部分回顾了两个相关问题。第六部分独立于其余部分，回顾了半群理论家称为素性的两个相关条件，并在群类别中讨论了这些条件与Tarski素性的关系。第七部分指出，当应用于任意变种中的有限代数时，其中一个条件的有趣特征化。本文提出了各种问题。
英文摘要：
If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a nonidentity element $p$ of $M_\mathcal{C}$ to be prime if, whenever it divides a product of two elements in that monoid, it divides one of them, and called an object of $\mathcal{C}$ prime if its isomorphism class has this property. McKenzie, McNulty and Taylor ask whether the category of nonempty semigroups has any prime objects. We show in section 2 that it does not. However, for the category of monoids, and some other subcategories of semigroups, we obtain examples of prime objects in sections 3-4. In section 5, two related questions open so far as I know, are recalled. In section 6, which can be read independently of the rest of this note, we recall two related conditions that are called primeness by semigroup theorists, and obtain results and examples on the relationships among those two conditions and Tarski's, in categories of groups. Section 7 notes an interesting characterization of one of those conditions when applied to finite algebras in an arbitrary variety. Various questions are raised.
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