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 consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid $R$. We endow a subset of $R$ with a \emph{left distributive monoid} structure and use it to extend the Dehornoy order on $B_{\infty}$ to an order on $R$. By using this order, we prove that $R$ is isomorphic to the monoid which is generated (geometrically) by shrinking braids.
Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be realized as distributive sets. The initial guess that any group may be embedded into Bin(X) for some X was complicated by an observation that if a binary operation is idempotent (a*a=a), then it commutes with every element of S. The first noncommutative subgroup of Bin(X), the symmetric group on three elements, was found in October of 2011 by Y.Berman. We show th…
We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a dec…
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…
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.
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.
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.
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.
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.
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
Institute of Natural Sciences and Mathematics, Ural Federal University（乌拉尔联邦大学自然科学与数学研究所）；Department of Mathematics, Nova Southeastern University（新南佛罗里达大学数学系）；School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
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).
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 …
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.
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…
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.
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.
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.
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…
Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We subsequently realise the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ as structure groups. We go on to obtain presentations of these groups via coherent categorifications of the varieties of higher-order associativity and of higher-order associativity and commutativity, respectively. These categorifications generalise Mac Lane's pen…
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…
