We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and counterexamples, also highlighting the particular role played by positive orderability. Moreover, we construct a non-left-orderable, positively right-orderable submonoid of the free product of the cyclic group of order 7 with the free group on 4 generators. Any group extension of that monoid has elements of order 7.

中文翻译：

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mono半群的右有序性与左有序性

我们研究了类半体的（总）左序和右序性之间的关系，特别是通过各种结构性观察和反例来说明有限情况，还强调了正序性所起的特殊作用。此外，我们构造了7阶循环组的自由乘积与4个生成器上的自由基的非左序，正右序次亚类。该monoid的任何组扩展都具有阶数为7的元素。