V. Yu. Shaprynskii, B. M. Vernikov
We completely determine all lower-modular elements of the lattice of all semigroup varieties. As a corollary, we show that a lower-modular element of this lattice is modular.
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We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.
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.
We completely determine all semigroup varieties satysfiyng a permutational identity of length 3 that are modular elements of the lattice of all semigroup varieties. Using this result, we provide an example of a semigroup variety that is a modular but not cancellable element of this lattice.
We completely determine all commutative semigroup varieties that are cancellable elements of the lattice SEM of all semigroup varieties. In particular, we prove that, for commutative varieties, the properties of being cancellable and modular elements of SEM are equivalent.
In this work, the lattice of varieties of semigroups and the lattice of varieties of I-semigroups (a common setting for both the variety of completely regular semigroups and the variety of inverse semigroups) are studied with respect to the following concepts: a variety V of (I-)semigroups is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many (distinct) K-classes.
We describe overcommutative varieties of semigroups whose lattice of overcommutative subvarieties satisfies a non-trivial identity or quasiidentity. These two properties turn out to be equivalent.
We completely determine all semigroup [epigroup] varieties that are cancellable elements of the lattice of all semigroup [respectively epigroup] varieties.
We completely determine all distributive, codistributive, standard, costandard, and neutral elements in the lattice of overcommutative semigroup varieties, thus correcting a gap contained in an earlier article by the second author.
Let M be a commutative monoid. We provide an explicit first-order formular that defines the variety generated by M in the lattice of commutative semigroup varieties.
The common behaviour of many families of numerical semigroups led up to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still out of these definitions. To overcome this situation, here we introduce the concept of restricted Frobenius variety (or R-variety). We will generalize most of the results for varieties and pseudo-varieties to R-varieties. In particular, we will study the tree structure that arise within them.
A semigroup variety V is said to be locally K-finite, where K stands for any of Green's relations H, R, L, D, or J, if every finitely generated semigroup from V has only finitely many K-classes. We characterize locally K-finite varieties of finite axiomatic rank in the language of "forbidden objects".
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 completely determine upper-modular, codistributive and costandard elements in the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being upper-modular and codistributive elements in the mentioned lattice are equivalent. Moreover, in the nil-case the properties of being elements of all three types turn out to be equivalent.
We study varieties of semigroups related to completely 0-simple semigroup. We present here an algorithmic descriptions of these varieties interms of "forbidden" semigroups.
In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-pres…
We study varieties of semigroups related to completely 0-simple semigroup. We present here an algorithmic descriptions of these varieties in terms of "forbidden" semigroups. We also describe residually completely 0-simple varieties of semigroups in terms of identities.
We survey results concerning special elements of nine types (modular, lower-modular, upper-modular, cancellable, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and certain its sublattices, mainly in the lattices of all commutative varieties and of all overcommutative ones. Several open questions are formulated. The work is regularly updated and modified as new results and/or articles appear.
In 2012, the second author introduced and examined a new type of algebras as a generalization of De Morgan algebras. These algebras are of type (2,0) with one binary and one nullary operation satisfying two certain specific identities. Such algebras are called implication zroupoids. They invesigated in a number of articles by the second author and J.M.Cornejo. In these articles several varieties of implication zroupoids satisfying the associative law appeared. Implication zroupoids satisfying the associative law are called implication semigroups. Here we completely describe the lattice of all …
Modular elements of the lattices of varieties of semigroups and epigroups. I
本文研究了 semigroup 和 epigroup 变种 lattice 中的模元素，提出了更严格的条件以确定变种是否为模元素，并为后续的完整分类奠定基础。
The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety COM of all commutative semigroups or a nil-variety or the join of a nil-variety with the variety of semilattices. Second, we prove that if a commutative nil-variety is a modular element of Com then it may be given within COM by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular w…
The aim of this paper is to provide an atlas of identity bases for varieties generated by small semigroups and groups. To help the working mathematician easily find information, we provide a companion website that runs in the background automated reasoning tools, finite model builders, and GAP, so that the user has an automatic \textit{intelligent} guide on the literature. This paper is mainly a survey of what is known about identity bases for semigroups or groups of small orders, and we also mend some gaps left unresolved by previous authors. For instance, we provide the first complete and ju…
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 study special elements of eight types (namely, neutral, standard, costandard, distributive, codistributive, modular, lower-modular and upper-modular elements) in the lattice EPI of all epigroup varieties. Neutral, standard, costandard, distributive and lower-modular elements are completely determined. A strong necessary condition and a sufficient condition for modular elements are found. Modular elements are completely classified within the class of commutative varieties, while codistributive and upper-modular elements are completely determined within the wider class of strongly permutative…
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 lattice of varieties of semigroups with completely regular square. II
本文在模群意义下求解满足 $xy = (xy)^n$ 的自由半群词问题，证明簇 ${\cal {SCR}}_n$ 不等于 ${\cal{CR}}_n$ 与 ${\cal{SI}}$ 的并，并刻画 ${\cal{SO}}_n$ 的格结构。
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 Lyapin's notebook: a collection of unsolved problems in Semigroup Theory
本文汇集了群论中未解决的关键问题，涵盖半群的潜在性质、嵌入性、结构问题及未来研究方向，纪念Lyapin并对学术界提供研究指引。
We examine varieties of epigroups as unary semigroups, that is semigroups equipped with an additional unary operation of pseudoinversion. The article contains two main results. The first of them indicates a countably infinite family of injective endomorphisms of the lattice of all epigroup varieties. An epigroup variety is said to be a variety of finite degree if all its nilsemigroups are nilpotent. The second result of the article provides a characterization of epigroup varieties of finite degree in a language of identities and in terms of minimal forbidden subvarieties. Note that the first r…
For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to $\overline{\mathbf H}$. Our characterization is effective whenever $\mathbf H$ has a decidable membership problem. In particular, the separation problem for $\overline{\mathbf H}$-languages is decidable for any decidable variety of finite groups $\mathbf H$. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.
Generalizing Lusztig's total positivity II : geometric properties
本文研究实半单李群中正半群的几何性质，引入旗流形的正部与非负部，并确定其拓扑，特别对辛旗流形给出显式描述。
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\leq n\leq 4$. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric Calabi-Yau equations are the fu…
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