monoid 的子集扩展
Subset expansions of monoids
AI总结：
研究 monoid 的子集扩展，探讨其代数性质及有限条件，揭示扩展 S(M) 的结构特征和性质提升条件。
摘要：
我们开始研究通过半直积得到的 monoid M 的扩展 S(M)，其中 M 自然地作用于其幂集的左侧（作为半格下的结构）。我们称这种扩展为 M 的 '子集扩展'。monoid S(M) 包含了几个广泛使用的 M 的扩展的图像，特别是前缀和 Szendrei 扩展（当 M 是自由时，这些 '较小' 的扩展在某些变种中产生自由代数）。我们首先关注代数性质，特别是由幂等元决定的性质。特别地，我们显示扩展 S 将群映射到适当的逆 monoid，单态 monoid 映射到适当的左限制 monoid，右可约 monoid 映射到左 ample monoid，右丰富 monoid 映射到右丰富 monoid，左可约 monoid 映射到右充分 monoid。随后，我们关注有限条件。我们检查了弱左一致性条件（每个有限生成的左理想都有一个有限的左 act 呈现）；相关的条件包括性质 (L)、左理想 Howson、有限左等同以及对应的左-右对偶概念。这些条件在 retract 下被保持，因此如果 S(M) 满足我们的一个有限条件，则 M 也必须满足，但反之则不成立。为了从 M 到 S(M) 提升一个性质，它必须被加强。事实上，我们显示 S(M) 满足性质 (L)（或其左-右对偶）当且仅当 M 是有限的。我们提供了 monoid M 的精确特征，使得 S(M) 是：左（或右）理想 Howson；有限左等同；以及（因此）弱左一致。我们给出了 S(M) 为有限右等同和因此弱右一致的充分条件。
英文摘要：
We initiate the study of the expansion $\mathcal{S}(M)$ of a monoid $M$ obtained via the semidirect product of $M$ acting naturally on the left of its power set (regarded as a semilattice under union). We term this the `subset expansion' of $M$. The monoid $\mathcal{S}(M)$ contains the images of several expansions of $M$ of wide interest and use in semigroup theory, in particular the prefix and Szendrei expansions (in the case where $M$ is free, these `smaller' expansions produce free algebras in certain varieties). We first focus on algebraic properties, specifically those determined by idempotents. Particularly, we show that the expansion $\mathcal{S}$ maps groups to proper inverse monoids, unipotent monoids to proper left restriction monoids, right cancellative monoids to left ample monoids, right abundant monoids to right abundant monoids, and left cancellative monoids to right adequate monoids. Subsequently, we focus on finitary conditions. We examine the condition of weak left coherence (every finitely generated left ideal has a finite presentation as a left act); the related conditions of property (L), left ideal Howson, finitely left equated, and each of the corresponding left-right dual notions. Each of these conditions is preserved under retract, from which it is immediate that if $\mathcal{S}(M)$ satisfies one of our finitary conditions, then so must $M$, but the converse is not true. For a property to `lift' from $M$ to $\mathcal{S}(M)$ it must undergo a strengthening. Indeed, we show that $\mathcal{S}(M)$ satisfies property (L) (or its left-right dual) if and only if $M$ is finite. We provide exact characterisations of the monoids $M$ such that $\mathcal{S}(M)$ is: left (or right) ideal Howson; finitely left equated; and (consequently) weakly left coherent. We give sufficient conditions for $\mathcal{S}(M)$ to be finitely right equated and hence weakly right coherent.
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