论文标题
解决问题的解决方案
Solution to a problem by FitzGerald
论文作者
论文摘要
菲茨杰拉德（Fitzgerald）确定了四个条件（RI），（ur），（ri*）和（ur*），它们必须由代数满足，如果其内态的单态性具有通勤的情况。我们表明，通过给出满足四种特性的代数的示例，这些条件是不够的，使得它的内态性很大，没有通勤的同性恋。 This settles a problem presented by Fitzgerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible.最后，我们列出了一些类别，其中菲茨杰拉德的问题具有肯定的答案。
FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra, if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by Fitzgerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald's question has an affirmative answer.
