Proper two-sided restriction semigroups and partial actions
Proper two-sided restriction semigroups and partial actions
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DOI：
10.1016/j.jpaa.2011.10.015
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发表时间：
2012-04
影响因子：
0.8
通讯作者：
Claire Cornock;V. Gould
中科院分区：
文献类型：
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作者：
Claire Cornock;V. Gould
双边限制半群和它们的手型有许多来源。最近吸引了大量的兴趣，他们出现在文献中的名称过多。左限制半群类本质上提供了部分映射半群的公理化。已知这类允许真覆盖，并且真左限制半群可以由作用在半格左的么半群来描述。任何真左限制半群都嵌入到半格的幺半群的半直积中，并且这个结果在左限制范畴的更广泛的背景下是已知的。对偶结果对右限制半群成立。关于双侧限制半群，我们能说些什么呢？当然，正确的封面是存在的。这里我们考虑真限制半群是否可以用作用在半格两边的么半群自然地描述。结果表明，为了得到真限制半群的全类，必须利用么半群的部分作用，从而恢复了Petrich和Reilly以及Lawson分别关于逆半群和富足半群的结果.我们还描述了一类真限制半群，使得部分作用可以相互扩展为作用。真逆和自由限制半群（它们是真的）具有这种形式，但我们给出了不具有这种形式的真限制半群的例子。
Two-sided restriction semigroups and their handed versions arise from a number of sources. Attracting a deal of recent interest, they appear under a plethora of names in the literature. The class of left restriction semigroups essentially provides an axiomatisation of semigroups of partial mappings. It is known that this class admits proper covers, and that proper left restriction semigroups can be described by monoids acting on the left of semilattices. Any proper left restriction semigroup embeds into a semidirect product of a semilattice by a monoid, and moreover, this result is known in the wider context of left restriction categories. The dual results hold for right restriction semigroups. What can we say about two-sided restriction semigroups, hereafter referred to simply as restriction semigroups? Certainly, proper covers are known to exist. Here we consider whether proper restriction semigroups can be described in a natural way by monoids acting on both sides of a semilattice. It transpires that to obtain the full class of proper restriction semigroups, we must use partial actions of monoids, thus recovering results of Petrich and Reilly and of Lawson for inverse semigroups and ample semigroups, respectively. We also describe the class of proper restriction semigroups such that the partial actions can be mutually extendable to actions. Proper inverse and free restriction semigroups (which are proper) have this form, but we give examples of proper restriction semigroups which do not.