源自左$E$-完备化的左限制幺半群
Left restriction monoids from left $E$-completions
AI总结：
该研究提出左$E$-完备化这一单侧幂等完备化构造，刻画其生成左限制半群的条件，并将其应用于分解三类半群结构，拓展了半群理论中幂等完备化的研究范畴。
摘要：
给定幺半群$S$，其中$E$是其幂等元的任意非空子集，我们提出一种新颖的单侧幂等完备化形式，称之为左$E$-完备化。一般而言，该构造会得到一种被Gould和Hollings称为“星座（constellation）”的小范畴的单侧变体。在特定条件下，该星座是归纳的，即其部分乘法可被拓展以生成左限制半群——这是一类一元半群，其一元运算用于刻画定义域。我们研究了满足该性质的配对$S,E$的性质，并刻画了那些可作为其“定义域为1的元素构成的子幺半群”的左$E$-完备化的左限制半群。作为初步应用，我们将集合$X$上部分函数构成的左限制半群、以及$X$上左满划分构成的右限制半群，分别分解为$X$上变换半群$T_X$的左$E$-完备化和右$E$-完备化；同时将$X$上在恶魔复合下的二元关系构成的左限制半群，分解为左满二元关系的左$E$-完备化。在包括这三个例子在内的诸多情形中，该构造可嵌入到半群的Zappa-Szép积中。
英文摘要：
Given a monoid $S$ with $E$ any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left $E$-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs $S,E$ for which this happens, and characterise those left restriction semigroups that arise as such left $E$-completions of their submonoid of elements having domain $1$. As first applications, we decompose the left restriction semigroup of partial functions on the set $X$ and the right restriction semigroup of left total partitions on $X$ as left and right $E$-completions respectively of the transformation semigroup $T_X$ on $X$, and decompose the left restriction semigroup of binary relations on $X$ under demonic composition as a left $E$-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szép product.
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