AI 中文总结
本文研究左$h$-埃雷斯曼半群，证明其自由对象与自由左$h$-恰当半群一致，给出其有向树描述并研究相关有限性条件。
AI 中文摘要
左$h$-恰当半群类介于左 ample 半群类与左埃雷斯曼半群类之间。相关类的自由对象可通过有向树描述，而自由左$h$-恰当半群此前仅能通过自由积的里斯商描述。本文重新引入（左）$h$-埃雷斯曼半群类，证明其自由对象与自由（左）$h$-恰当半群一致，并给出该类自由对象的有向树描述；利用该新描述研究这类结构的若干性质，包括近期受关注的有限性条件。
英文摘要
The class of left $h$-adequate semigroups sits between the classes of left ample and left Ehresmann semigroups. Whilst free objects in related classes have descriptions in terms of directed trees, free left $h$-adequate semigroups have yet only been described in terms of Rees quotients of free products. Here, we reintroduce the class of (left) $h$-Ehresmann semigroups, show their free objects coincide with free (left) $h$-adequate semigroups, and give a description of these free objects in terms of directed trees. We use our new description in investigating various properties of these structures, including finitary conditions of recent interest.
Comments22 pages, 6 figures
