Ehresmann monoids
Ehresmann monoids
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DOI：
10.1016/j.jalgebra.2015.06.035
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发表时间：
2014
影响因子：
9
通讯作者：
Mário J. J. Branco-Mário-J.-J.-Branco-2070290661;Gracinda M. S. Gomes;Victoria Gould
中科院分区：
文献类型：
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作者：
Mário J. J. Branco-Mário-J.-J.-Branco-2070290661;Gracinda M. S. Gomes;Victoria Gould
Ehresmann幺半群形成了各种各样的双一元幺半群，也就是说，幺半群配备了两个基本的一元运算，其图像重合并形成投影的半格。在任何集合X上的二元关系B X的幺半群具有定义域和值域的一元运算是Ehresmann。逆幺半群，通过瓦格纳-普雷斯顿表示定理被认为是B X的二元子幺半群，因此也是Ehresmann。在另一个极端，任何幺半群都是Ehresmann，其中一元运算将所有元素带到幺半群单位元。我们证明在这里使用半格和幺半群作为积木，埃雷斯曼幺半群有一个丰富的结构，从根本上不同于逆幺半群，事实上，从临时类的限制幺半群。本文介绍了一个概念的适当性Ehresmann幺半群，严格控制结构，并依赖于集的发电机。我们证明了如何通过保序映射从一个两边都被幺半群T作用的半格Y构造一个满足适当性条件的Ehresmann幺半群P（T，Y）。证明了X上的自由Ehresmann么半群具有形式P（X ∈，Y）.下一个问题是关于真覆盖的存在性。我们以肯定的方式回答它，证明了任何Ehresmann么半群M都有形式P（X ∈，E）的覆盖，其中E是M的投影半格。这里的“cover”是一个态射下的原像，它分隔E中的元素。
Ehresmann monoids form a variety of bi-unary monoids, that is, monoids equipped with two basic unary operations, the images of which coincide and form a semilattice of projections. The monoid of binary relations B X on any set X with unary operations of domain and range is Ehresmann. Inverse monoids, regarded as bi-unary submonoids of B X via the Wagner–Preston representation theorem, are therefore also Ehresmann. At the other extreme, any monoid is Ehresmann, where the unary operations take all elements to the monoid identity. We demonstrate here using semilattices and monoids as building blocks that Ehresmann monoids have a rich structure, fundamentally different from that of inverse monoids and, indeed, from that of the interim class of restriction monoids. The article introduces a notion of properness for Ehresmann monoids, that tightly controls structure and is dependent upon sets of generators. We show how to construct an Ehresmann monoid P (T, Y) satisfying our properness condition from a semilattice Y acted upon on both sides by a monoid T via order preserving maps. The free Ehresmann monoid on X is proven to be of the form P (X⁎, Y). The next question deals with the existence of proper covers. We answer it in a positive way, proving that any Ehresmann monoid M admits a cover of the form P (X⁎, E), where E is the semilattice of projections of M. Here a ‘cover’is a preimage under a morphism that separates elements in E.