Sergey V. Gusev, Edmond W. H. Lee, Boris M. Vernikov
We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated.
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Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascen…
On the lattice of overcommutative varieties of monoids
本文证明任意有限格都是幺半群超交换簇格的子格的同态像，从而否定回答了所有幺半群簇的格是否满足非平凡恒等式的问题。
Varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive
本文完全分类了所有具有中心幂等元的非周期幺半群簇，并刻画了其中子簇格为分配格的簇。
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. We present a new pair of limit varieties of monoids and show that together with the five limit varieties of monoids previously discovered by Jackson, Zhang and Luo and the first-named author, there are exactly seven limit varieties of J-trivial monoids.
We completely determine all varieties of monoids on whose free objects all fully invariant congruences or all fully invariant congruences contained in the least semilattice congruence permute. Along the way, we find several new monoid varieties with the distributive subvariety lattice (only a few examples of varieties with such a property are known so far).
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
Definability of the variety generated by a commutative monoid in the lattice of commutative semigroup varieties
设M是交换幺半群。我们提供了一个明确的一阶公式，定义了由M在交换半群簇的格中生成的簇。
We completely classify all standard elements in the lattice of all monoid varieties. In particular, we prove that an element of this lattice is standard if and only if it is neutral.
Varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is distributive
本文研究了无周期单oids中幂等元互相交换且子变种格为分配的种类，进行了完整分类。
We show that limit varieties of monoids recently discovered by Gusev, Zhang and Luo and their subvarieties are generated by monoids of the form $M_τ(W)$ for certain congruences $τ$ on the free monoid. The construction $M_τ(W)$ is a generalization of widely used Dilworth-Perkins construction. Using this construction, we find explicit generators for Gusev limit varieties and give a short reproof to the fact that Zhang-Luo limit variety is non-finitely based.
Varieties of monoids with a distributive subvariety lattice
本文完全分类了所有子变体格为分配的无周期幺半群变体，其中无周期指所有子群平凡。
The sets of all neutral, distributive and lower-modular elements of the lattice of semigroup varieties are finite, countably infinite and uncountably infinite, respectively. In 2018, we established that there are precisely three neutral elements of the lattice of monoid varieties. In the present work, it is shown that the neutrality, distributivity and lower-modularity coincide in the lattice of monoid varieties. Thus, there are precisely three distributive and lower-modular elements of this lattice.
We completely classify all neutral or costandard elements in the lattice $\mathbb{MON}$ of all monoid varieties. Further, we prove that an arbitrary upper-modular element of $\mathbb{MON}$ except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of $\mathbb{MON}$. Thus, the problems of describing codistributive or upper-modular elements of $\mathbb{MON}$ are completely reduced to the completely regular case.
Characterization of Cross varieties of $J$-trivial monoids
本文研究了$J$-平凡单oid变种的交叉性质，证明其为交叉变种当且仅当排除特定的14个几乎交叉变种。
Cross varieties of aperiodic monoids with commuting idempotents
本文分类了具有交换幂等元的非周期幺半群中所有满足有限基、有限生成且子簇有限的Cross簇。
Cross varieties of aperiodic monoids
本文研究非周期幺半群的交叉簇，证明其为交叉簇等价于排除22个近交叉簇，且该列表穷尽了所有此类近交叉簇。
A variety of universal algebras is called a chain variety if its subvariety lattice is a chain. Non-group chain varieties of semigroups were completely classified by Sukhanov in 1982. Here we completely determine non-group chain varieties of monoids as algebras of tyoe (2,0).
Minimal monoids generating varieties with complex subvariety lattices
该研究证明6元素Brandt幺半群生成最小的有限泛幺半群簇，且两个子簇格分别为6、7元素链的Cross幺半群簇的并可构成含不可数多子簇的有限泛簇，推进了幺半群簇子簇格复杂性的研究。
A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there exists a finitely universal variety of monoids. The main objective of the present article is to exhibit the first examples of finitely universal varieties of monoids. The finite universality of these varieties is established by showing that the lattice of equivalence relations on every sufficiently large finite set is anti-isomorphic to some subinterval of …
Limit varieties generated by finite non-J-trivial aperiodic monoids
针对Jackson与Lee提出的六元幺半群生成的遗传有限基簇$\boldsymbol{\rm E}^1$，研究者确定其有限生成子簇的句法幺半群，发现其中一个有限幺半群与某七元幺半群可生成新的极限簇。
In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.
Modular elements in the lattice of monoid varieties
本文研究了单侧范畴格中模元素的分类，通过分析五元素子格的结构确定其模性质。
The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice $\mathbb{MON}$ of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in $\mathbb{MON}$ is established. These results play a crucial role in the complete description of all cancellable elements of the lattice $\…
A limit variety is a variety that is minimal with respect to being non-finitely based. The two limit varieties of Marcel Jackson are the only known examples of limit varieties of aperiodic monoids. Our previous work had shown that there exists a limit subvariety of aperiodic monoids that is different from Marcel Jackson's limit varieties. In this paper, we introduce a new limit variety of aperiodic monoids.
Finiteness conditions for lattices of monoid varieties
本文分类了子簇格有限、满足降链条件或升链条件的含中心幂等元非周期幺半群簇，并证明该类中有限子簇格与升链条件等价，且升链条件蕴含降链条件。
Limit varieties of monoids satisfying a certain identity
本文解决了非周期幺半群极限簇分类中的一个关键子问题，证明满足恒等式 $xsxt \approx xsxtx$ 的极限簇恰好有两个，并由此推出阶数不超过五的幺半群生成的簇至多含有可数多个子簇。
Small monoids generating varieties with uncountably many subvarieties
本文证明十阶 Rees 商幺半群 M(aabb) 生成具有不可数多个子簇的簇，肯定回答了 Glasson 的问题，并由此构造出第一个极小且有限基的六阶类型 2^{ℵ₀} 幺半群。
Limit varieties of aperiodic monoids
本文研究了非周期单oid的极限变种，提出一个新的极限变种，并证明其他可能的极限变种包含在特定变种的联合中。
Semigroup identities and varieties of plactic monoids
本文研究有限秩 plactic 幺半群的半群恒等式，给出 n≥4 时更短的新恒等式，并据此证明不同秩的 plactic 幺半群生成互不相同的半群簇。
The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green's relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same $\mathcal{J}$-class and their multipl…
Coherent presentations of structure monoids and the Higman-Thompson groups
结构幺半群和群是方程簇的代数不变式。我们将展示如何构建这些对象的演示文稿从连贯的equational品种，概括Dehornoy的几个结果。我们随后实现了更高的Thompson群$F_{n,1}$和Higman-Thompson群$G_{n,1}$作为结构群。我们继续通过高阶结合性和高阶结合性与交换性的各种相干的简化来得到这些群的表示。这些简化概括了Mac Lane的五边形和六边形条件的连贯关联和交换双函子。
On Algebraic Semigroups and Monoids
我们提出了一些基本结果（可能非线性）代数半群和幺半群。这些包括一个版本的Chevalley的结构定理的不可约代数monoid，并描述所有代数半群结构的曲线和完整的品种。
A finite interval in the subsemigroup lattice of the full transformation monoid
本文描述了\emph{full transformation semigroup} $Ω^Ω$的子半群格的一部分，它由可数无限集$Ω$上的所有映射组成. Gavrilov证明了$Ω^Ω$中存在5个包含对称群$\sym(Ω)$的极大子半群.我们所描述的$Ω^Ω$的子半群格的部分是这五个极大子半群与$Ω^Ω$的交集之间的部分。我们证明了在这个区间上只有38个子半群，而不是$\sym(Ω)$和$Ω^Ω$之间的$2^{2^{\aleph_0}}$子半群.
Partial mirror symmetry, lattice presentations and algebraic monoids
这是第二次在一系列的文件，发展理论的反射monoid，动机理论的反射群。反射幺半群在arXiv：0812.2789中首次引入。在本文中，我们研究他们的表现为抽象幺半群。一路上，我们也发现一般介绍某些联合半格（作为monoid下加入），我们解释了两个特殊类别的例子：面对凸多胞体和几何格，特别是交叉格超平面安排。另一个副产品是一般介绍伦纳幺半群的代数幺半群，我们说明在特殊情况下的“经典”代数幺半群。
Normal submonoids and congruences on a monoid
该研究引入幺半群的正规子幺半群概念，证明其构成完全格并计算了有限全变换幺半群的对应格，提出通过正规子幺半群与商的单位同余计算幺半群同余格的新策略，为相关经典计算提供新视角。
We completely determine all semigroup varieties satisfiyng a permutational identity of length 3 that are cancellable elements of the lattice of all semigroup varieties. Using this result, we provide a series of new examples of semigroup varieties that are a modular but not cancellable elements of this lattice.
Identities and quasiidentities in the lattice of overcommutative semigroup varieties
我们描述了半群的超交换簇，其超交换子簇的格满足非平凡恒等式或拟恒等式。这两个属性是等价的。
Lower-modular elements of the lattice of semigroup varieties. III
我们完全确定了所有半群簇的格的所有下模元素。作为推论，我们证明了这个格的一个低模元是模的。
We completely determine all commutative epigroup varieties that are cancellable elements of the lattice EPI of all epigroup varieties. In particular, we verify that a commutative epigroup variety is a cancellable element of the lattice EPI if and only if it is a modular element of this lattice.
On modular and cancellable elements of the lattice of semigroup varieties
我们完全确定了所有满足长度为3的置换恒等式的半群簇是所有半群簇的格的模元素。利用这个结果，我们提供了一个例子，半群品种是一个模块化的，但不能取消这个格的元素。
Cancellable elements of the lattice of semigroup varieties
我们完全确定了所有交换半群簇，这些交换半群簇是所有半群簇的格SEM的可消元。特别是，我们证明了，交换品种，是可撤销的性质和模元素的SEM是等价的。
Special elements in the lattice of overcommutative semigroup varieties revisited
我们完全确定了超交换半群簇格中的所有分配元、协分配元、标准元、协标准元和中性元，从而纠正了第二作者在前一篇文章中的一个缺陷。
Tamari Lattices and the symmetric Thompson monoid
我们调查之间的联系Tamari格和汤普森群F，总结的事实是，F是一组分数的某一monoid F+sym的凯莱图包括所有Tamari格。在这种对应关系下，Tamari格运算是F+sym中的最小公倍数和最大公因数运算的对应运算。作为一个应用，我们证明了对于任意n，在第n个Tamari格中存在一个长度为l的链，其端点的距离不超过12 l/n.
A lattice-ordered monoid on multilayer networks
本文在多层网络上引入格序偏幺半群结构，通过研究保序映射描述序结构，定义并推导了格序幺半群的主要性质。
Tamari lattices, forests and Thompson monoids
给出了将对称群上的Tamari格看作弱Bruhat阶格与Thompson群F的正幺半群P之间的联系. Tamari同余类对应于P中的等价元素类。P中的两个著名的标准形对应于确定Tamari类的弱Bruhat序的区间的端点。在幺半群P中，这些对应于字典上最大和字典上最小的形式，而在排列水平上，它们对应于132-避免和231-避免排列。森林自然出现在这两种情况下，因为它们被用来模拟汤普森幺半群的排列和元素。然后将这种联系推广到（（k-1）n+2）-半群分拆为（k+1）-半群和Thompson幺半群P_k，k >1上的Tamari序.
We completely determine all semigroup [epigroup] varieties that are cancellable elements of the lattice of all semigroup [respectively epigroup] varieties.
We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.
Upper-modular and related elements of the lattice of commutative semigroup varieties
完全确定了所有交换半群簇的格中的上模元、余分配元和余标准元。特别地，我们证明了在所提到的格是上模和共分配元素的性质是等价的。此外，在nil情况下，成为所有三种类型的元素的属性是等价的。
We completely determine all cancellable elements in the lattice OC of overcommutative semigroup varieties. In particular, we prove that an overcommutative semigroup variety is a cancellable element of the lattice OC if and only if it is a neutral element of this lattice
We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and j…
For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We sh…
Heights of one- and two-sided congruence lattices of semigroups
受对称群与全变换幺半群格高度问题启发，提出计算半群单侧与双侧同余格高度的通用方法，并推导出多种变换、矩阵与分拆幺半群的精确高度公式。
The lattice of submonoids of the uniform block permutations containing the symmetric group
本文研究均匀块置换幺半群中含对称群的子幺半群格，借助整数分拆上的新偏序下集证明其分配性，并给出 \mathscr{J}-类大小与不可约模维数平方和的关系。
Braid groups of type ADE, Garside monoids, and the categorified root lattice
通过$\mathbb{B}_W$在分次投射Z字形模的同伦范畴$\mathcal{K}$上的作用（它把Weyl群$W$在根格上的作用归为范畴），研究了ADE型Artin-Tits辫群$\mathbb{B}_W$.在Brav-Thomas的基础上，我们定义了由$\mathcal{K}$上的规范$t$ -结构诱导的$\mathbb{B}_W$上的一个度量，并证明了这个度量与辫子群的标准正幺半群$\mathbb{B}_W^+$的规范生成元中的字长度量一致.我们还定义了，为每一个选择的Coxeter元素$c$在$W$，一个baric结构上的$\mathcal{K}$。我们使用这些baric结构来定义指标的辫子群，我们确定这些指标的字长指标的Birman-Ko-Lee/Bessis双发电机相关的双正幺半群$\mathbb{B}_{W.c}^\vee$。作为结果，我们给出了标准和对偶正幺半群注入群的新证明，给出了标准和对偶正幺半群中成员问题的线性代数解，并给出了$\mathbb{B}_W$作用于$\mathcal{K}$的忠实性的新证明.最后，我们利用$\mathcal{K}$上的baric和$t$ -结构的相容性证明了Digne和Gobet关于ADE辫群的对偶单生成元的规范字长的一个猜想.
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate a…
We study the algebraic properties of binary relations whose underlying digraph is smooth, that is has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under defin…
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