Cross varieties of aperiodic monoids with commuting idempotents
S. V. Gusev
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。
一个代数簇被称为Cross簇，如果它是有限基的、有限生成的，并且具有有限多个子簇。在本文中，我们分类了所有具有交换幂等元的非周期幺半群的Cross簇。
A variety of algebras is called Cross if it is finitely based, finitely generated, and has finitely many subvarieties. In present article, we classify all Cross varieties of aperiodic monoids with commuting idempotents.
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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. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
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。
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.
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。
We characterise the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated to certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.
Subgroups of free idempotent generated semigroups: full linear monoid
发展了一些新的拓扑工具来研究自由幂等生成半群的极大子群。作为应用，我们证明了除环Q上n × n矩阵（n>2）的满矩阵幺半群的双序集的自由幂等生成半群的秩1分支有极大子群同构于Q的乘法子群.
A note on free idempotent generated semigroups over the full monoid of partial transformations
最近，Gray和Ruskuc（arXiv：1101.1833）证明了，如果e是集合{1，…，n}到自身且k<=n-2，则含e的全变换幺半群T_n上的自由幂等元生成半群的极大子群同构于对称群S_k。我们证明了当T_n被{1，…，n}。
Maximal subgroups of free idempotent generated semigroups over the full transformation monoid
设T_n是集合{1，…，n}的组合。设E = E（T_n）表示T_n的幂等元集，设e是满足|im（e）|=r < n-1。证明了E上含有e的自由幂等元生成半群的极大子群同构于对称群Sr.
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。
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
Maximal subgroups of free idempotent generated semigroups over the full linear monoid
证明了除环Q上n × n矩阵的全线性幺半群的双序集的自由幂等元生成半群的秩r分支有同构于一般线性群GL_r（Q）的极大子群，其中n和r是正整数，r < n/3.
Aperiodic Flows on Finite Semigroups II: Smallish Monoids Suffice for Complexity 1
本文研究了有限半群的非周期流性质，通过构造性方法将任意有限半群嵌入到小规模单体的评估半群中，并利用流理论证明了群映射半群具有非周期流当且仅当对应的评估半群也具有非周期流，从而将Krohn-Rhodes复杂度1的计算简化到小规模单体类中。
We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempo…
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。
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。
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs. We show that each proper ideal of the partition monoid P_n is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and …
With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, et…
Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1…
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrar…
We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated su…
We investigate the structure of the twisted Brauer monoid $\mathcal B_n^τ$, comparing and contrasting it to the structure of the (untwisted) Brauer monoid $\mathcal B_n$. We characterise Green's relations and pre-orders on $\mathcal B_n^τ$, describe the lattice of ideals, and give necessary and sufficient conditions for an ideal to be idempotent-generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent-generated, its…
We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for $ω$-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free…
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…
On the question of embedding a semigroup into an idempotent generated one
本文利用半直积构造给出了半群到深度为4的半带（幂等元生成半群）的一种新的嵌入.我们的嵌入不假设S是幺半群（尽管它假设了一个较弱的条件），并且也适用于（非幺半群）正则半群。事实上，这个半直积对正则半群特别有用，因为我们可以定义这些半群的另一个嵌入到深度为2的较小半带中。然后我们将把我们的构造与其他已知的嵌入进行比较，我们将看到S的某些性质被我们的嵌入所保持。
From free idempotent monoids to free multiplicatively idempotent rigs
本文研究乘法幂等半环（mirig），证明有限生成元上的自由 mirig 是有限的并计算其大小，工作源于在 Mastodon 上计算两个生成元自由 mirig 规模的合作。
Nilpotent groups, solvable groups and factorizable inverse monoids
本文引入逆半群中的次中心与中心幂等元序列，证明了可因子化逆幺半群中此类序列的同构性（类比Jordan-Hölder定理），并据此定义了G-幂零与G-可解逆幺半群，为研究幂零群和可解群提供了新途径。
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…
Free idempotent generated semigroups and endomorphism monoids of free $G$-acts
对双序集$E$上的自由幂等生成半群$\mathrm{IG}(E)$的研究始于20世纪70年代Nambooripad的开创性工作，最近又出现了一些新的方法，包括几何方法和组合方法。本文研究了$\mathrm{IG}(E)$在$E$是环积$G\wr \mathcal{T}_n$的双序集的情况下的性质，其中$G$是群，$\mathcal{T}_n$是$n$元素上的全变换幺半群.这个环积同构于$n$生成元上的自由$G$ -act $F_n(G)$的自同态幺半群，这为我们提供了一个方便的途径。我们说$F_n(G)$的元素的秩是它的图像中（自由）生成器的最小数量。令$\varepsilon=\varepsilon^2\in F_n(G).$由于相当直接的原因，已知如果$\mathrm{rank}\,\varepsilon =n-1$（分别为$n$），则包含$\varepsilon$的$\mathrm{IG}(E)$的极大子群是自由的（分别为平凡的）。我们证明了如果$\mathrm{rank}\,\varepsilon =r$，其中$1\leq r\leq n-2$，那么包含$\varepsilon$的$\mathrm{IG}(E)$的极大子群同构于$F_n(G)$中的极大子群，从而同构于$G\wr \mathcal{S}_r$，其中$\mathcal{S}_r$是$r$元素上的对称群。我们之前…
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
We show that every finite inverse monoid has an idempotent-separating cover by a finite F-inverse monoid. This provides a positive answer to a conjecture of Henckell and Rhodes.
We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and stro…
Presentation of monoids generated by a projection and an involution
该研究针对由对合和幂等元生成的strict 2-PIMs幺半群子问题，证明有限情形下可由单个额外等式生成，完成等式形式分类，并将Kuratowski定理纳入特例，为二阶元素生成的幺半群分类提供基础。
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。
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new …
We present a general method for proving that a semigroup is non-finitely based. The method is strong enough to cover the non-finite basis arguments in articles [1,3,4,5,7,8, 11,14,16,21,27,31,36,37]. In particular, the method allows to generalize the results in [1,8,36,37] and to simplify their proofs. The method also allows to remove one of the requirements on the "special system of identities" used by P. Perkins in [16] to find the first two examples of finite non-finitely based semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is non-finitely based…
Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids
研究投影代数\(P\)上自由投影生成的正则\(*\)-半群\(PG(P)\)及其与自由幂等生成半群\(IG(E)\)极大子群的关系，通过生成元和关系得到表示并应用于划分幺半群\(P_n\)，明确计算出相关极大子群。
We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that $S_n$ consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, and compute nilpotency indexes. Furthermore, we give a necessary and sufficient condition for the $j$…
We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if $\mathcal{C}$ is a reversal-closed super-$\operatorname{AFL}$, as defined by Greibach, then $M$ has word problem in $\mathcal{C}$ if and only if its compressed left monoid $L(M)$ has word problem in $\mathcal{C}$. As a special case, we may take $\mathcal{C}$ to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finall…
For a monoid $M$ and a subsemigroup $S$ of the full transformation semigroup $T_n$, the wreath product $M\wr S$ is defined to be the semidirect product $M^n\rtimes S$, with the coordinatewise action of $S$ on $M^n$. The full wreath product $M\wr T_n$ is isomorphic to the endomorphism monoid of the free $M$-act on $n$ generators. Here, we are particularly interested in the case that $S=Sing_n$ is the singular part of $T_n$, consisting of all non-invertible transformations. Our main results are presentations for $M\wr Sing_n$ in terms of certain natural generating sets, and we prove these via ge…
Left restriction monoids from left $E$-completions
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。
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。
The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups
本文为满足广义右ample条件的约化$E$-Fountain半群构造零态射范畴，并证明半群代数与收缩范畴代数同构，统一推广多项已有结果且适用于保序函数幺半群等新例子。
We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.
We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, $\mathscr J$-trivial and $\mathscr R$-trivial monoids and left regular bands of groups.
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and is contained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ is non-finitely based. We establish this result by analyzing $τ$-terms for $M$ where $τ$ is certain non-trivial congruence on the free semigroup, that is, we…
Let $m,n\in \mathbb{N}$, and $V$ be a $m$-dimensional vector space over a field $F$ of characteristic $0$. Let $U=F\oplus V$ and $R_n$ be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of $U^{\otimes n}$ in $FR_n$, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of $U^{\otimes n}$ in $FR_n$.
Infinite partition monoids
设$\mathcal P_X$和$\mathcal S_X$是无限集$X$上的划分幺半群和对称群.我们表明，$\mathcal P_X$可能产生的$\mathcal S_X$连同两个（但不少于）额外的分区，我们分类的$α,β\in\mathcal P_X$对$\mathcal P_X$产生的$\mathcal S_X\cup\{α,β\}$。我们还表明，$\mathcal P_X$可以产生的集合$\mathcal E_X$的所有幂等分区连同两个（但不少于）额外的分区。事实上，$\mathcal P_X$由$\mathcal E_X\cup\{α,β\}$生成当且仅当它由$\mathcal E_X\cup\mathcal S_X\cup\{α,β\}$生成。我们还对由$\mathcal E_X\cup\{α,β\}$生成$\mathcal P_X$的对$α,β\in\mathcal P_X$进行分类。在其他的结果中，我们证明了任何可数子集的$\mathcal P_X$包含在一个$4$生成的子半群的$\mathcal P_X$，和长度函数的$\mathcal P_X$是有界的任何生成集。
We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to …
On homological finiteness properties and free inverse monoids
基于幂等格作用构造充分条件，证明自由逆幺半群非FP_2型，并刻画FIM(1)的有限生成子幺半群为FP_2型当且仅当有限表示。
The modular representation theory of monoids
本文建立有限幺半群模表示论基础框架，引入分解矩阵与模特征，证明非模正特征表示可提升至特征零，并给出 Glover 定理新证明，同时研究 Cartan 矩阵非奇异性在特征零与正特征间的关系，提出统一新猜想。
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