New Trends in Algebras and Combinatorics

The following sections are included:

• Preface
• Contents

Professor Leonid Bokut was born in 1937 in a small town Krupki located in the Republic of Belarus which had been a part of the Soviet Union at that time. In 1953, he attended Moscow State University (MSU), Department of Mechanics and Mathematics. His supervisor at MSU was A.I. Shirshov, who later became one of the founders of the Siberian School of Algebra and Logics together with A.I. Malcev and M.I. Kargapolov…

This paper is about my life in mathematics, including my supervisors, my students and my main results. I will introduce them mainly in chronological order.

“This short paper [A. I. Shirshov, Some algorithmic problems for Lie algebras. Sib. Mat. Zh. 3, 292-296 (1962) – L.B.] describes work similar to that appearing in B. Buchbergers 1965 thesis inventing Gröbner bases, but in the context of Lie algebras. Preceding Buchberger by only three years, this paper, along with the two cited references, are the original papers defining what has become known as Gröbner–Shirshov bases.” This is a quotation by Michael Abramson, together with Rebecca Abramson the translator of the first English translation of the paper under review, cf. [A.I. Shirshov, “Certain algorithmic problems for Lie algebras,” SIGSAM Bull. 33, No. 2, 3-6 (1999; Zbl 1097.17502)]. Incidentally, there is another translation available in [L. A. Bokut, V. Latyshev, I. Shestakov, E. Zelmanov (editors), Selected works of A.I. Shirshov. Translated by Murray Bremner and Mikhail V. Kotchetov. Contemporary Mathematicians. Basel: Birkhäuser (2009; Zbl 1188.01028)]…

A cluttered ordering is a type of a cyclic ordering of a graph. It can provide minimize the number of disk operations. Mueller et al. (2005) decomposed the complete bipartite graph into isomorphic copies of the special bipartite graph H(h; t), where h and t are positive integers. They presented some edge labeling of the infinite families of H(1; t), H(2; t). Adachi (2017) defined the special bipartite graph H(h, k; t), which is extension of H(h; t), and presented some edge labeling of the infinite families of H(1, 2; t). In this paper, we investigate the case of H(1, k; t), and obtain the corresponding cluttered orderings.

The paper presents a method for recognition of polynomial completeness of n-ary finite quasigroup given by its Latin square. The method uses properties of the group G(Q). There is also given a construction of ternary finite quasigroups whose all binary deducts are polynomially complete.

Let R be a ring and σ an endomorphism of R. We recall that R is called an (S, 1)-ring if for a, b ∈ R, ab = 0 implies aRb = 0. We involve σ to generalize this notion. We say that R is a left σ-(S, 1) ring if for a, b ∈ R, ab = 0 implies aRb = 0 and σ(a)Rb = 0. We say that R is a right σ-(S, 1) ring if for a, b ∈ R, ab = 0 implies aRb = 0 and aRσ(b) = 0. R is called a σ-(S, 1) ring if it is both right and left σ-(S, 1) ring. In this paper we give examples of such rings and a relation between σ-(S, 1) rings, 2-primal rings, and σ(⁎)-rings.

We show that a certain class of matrix rings, with suitable endomorphisms σ are left σ-(S, 1) but not right σ-(S, 1), and vice versa.

Applying Gröbner—Shirshov technique, we prove that any post-Lie algebra injectively embeds into its universal enveloping postassociative algebra.

The left GC-lpp semigroups are known as the common generalizations of left ample semigroups and right inverse semigroups. In this paper, we will prove that the class of left GC-lpp semigroups is a quasivarity. Furthermore, the structure of free left GC-lpp semigroups will be described and investigated. The main result in this paper is that any left GC-lpp monoid has a F-cover.

In this paper, we give a survey concerning the influence and the related progress of the cover-avoiding subgroups on the structure of finite groups.

We compute the length function of the group algebra of quaternions over an arbitrary field.

The following sections are included:

• Introduction
• t-intersecting families
• Largest union-intersecting families and related problems
• Two- or more-part intersecting families
• Minimum shadows of t-intersecting families
• References

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polyhedrons and graphs.

The following sections are included:

• Introduction
• q-Analogs of Numbers
• q-Pochhammer Symbol
• q-Analogs and Vector spaces
• q-Weyl Calculus
• q-Binomial Theorem
• q-Derivative or Jackson’s Derivative
• q-Binomial Theorem
• q-Trigonometry
• Jackson’s Integral
• Jacobi Elliptic Functions & Theta Functions
• References

The continued fractions algorithm for isolating the real roots of polynomials with certainty is one of the most efficient known and widely used. It can be viewed as exploring a tree created by a sequence of simple transformations. In this paper we produce new upper bounds for the breadth of the tree that are significantly smaller than the degree of the input polynomial. We also consider the expected breadth under a reasonable distribution and derive a bound, subject to a plausible assumption, that grows logarithmically with the degree and coefficient size.

We demonstrate that the combinatorial rank of the multiparameter version of the Lusztig “small” quantum group u_q(𝔰𝔭_2n), or, equivalently, of the Frobenius-Lusztig kernel of type C_n, equals $\lfloor {\log }_2\left(n-1\right)\rfloor +2$ provided that q has a finite multiplicative order t > 3. It is known that the combinatorial rank of the multiparameter version of the Frobenius-Lusztig kernel of type A_n equals $\lfloor {\log }_{2}n\rfloor +1$, whereas in case B_n it is equal to $\lfloor {\log }_2\left(n-1\right)\rfloor +2$, and in case D_n it is $\lfloor {\log }_2\left(n-2\right)\rfloor +2$.

We present an approach to the computation of confluent systems of defining relations in associative conformal algebras based on the similar technique for modules over ordinary associative algebras.

In Ref. 6, the author has constructed natural maps from fundamental groups of topological spaces (restricted configuration spaces) to the groups $G_n^k$. In the present paper, we show that in the case of n = k+1, the group $G_{k+1}^k$ is isomorphic to the fundamental group of some (quotient space of) some configuration space up to finite index subgroups. In particular, this leads to the solution of word and conjugacy problems in $G_4^3$ and sheds light on $G_{k+1}^k$ for higher k.

This paper surveys some recent developments in the theory of presentations of inverse semigroups by generators and relations. Particular attention is paid to the word problem for one-relator inverse monoids and for inverse semigroups admitting a cycle-free presentation. Some applications of the theory to operator algebras and topology are discussed.

We consider the notions of the replicators, including the duplicator and triplicator, of a binary operad. As in the previous notions of di-Var-algebra and tri-Var-algebra in Ref. 14, they provide a general operadic definition for the recent constructions of replicating the operations of algebraic structures. We show that taking replicators is in Koszul dual to taking successors in Ref. 3 for binary quadratic operads and is equivalent to taking the white product with certain operads such as Perm. We also relate the replicators to the actions of average operators.

In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.

A right module M_R over a ring R is called a V-module if every simple right R-module is M-injective. A module N_R is defined to be hereditary if every submodule of N is projective. With these concepts, the main result of Ref. 6 can be restated in the form: a V-module P_R over a prime right Noetherian ring R is hereditary if and only if R_R is hereditary and P is R-projective.

In this paper we consider a general case for a right Noetherian right nonsingular ring R having a nonsingular V-module P_R, and show that all uniform right ideal U ⊆ R which are embedded in P_R generate a nonzero two-sided ideal 𝒰_R(P) ⊆ R such that 𝒰_R(P)_R is a V-module. Further, we show that, if 𝒰_R(P)_R is hereditary, then P_R is hereditary if and only if P is 𝒰_R(P)_P -projective (Theorem 3.1). If R is a right Noetherian right hereditary ring, then all R-projective V-module P_R are hereditary (Corollary 3.1).

If R is semisimple right Noetherian ring, then a V-module P_R is hereditary if and only if 𝒰_R(P)_R is hereditary and P_R is 𝒰_R(P)_R-projective. In this case, 𝒰_R(P) is a ring-direct summand of R (Theorem 2.1). There are known examples of prime Noetherian V-rings which are not hereditary. Over these rings, there are no nonzero hereditary modules.

The paper presents an elementary and unified approach to vector spaces over fields of order greater than or equal to one (the latter reducing to sets), based on three key principles. Firstly, use of quasigroups enables the field concept to be redefined in a way that admits a field of order one. Secondly, use of hyperquasigroups provides a recursive definition of linear combination that applies equally well to vector spaces over fields of order greater than or equal to one. Thirdly, it is recognized that relations rather than functions provide the correct morphisms for a category of sets to behave like categories of vector spaces over fields of order greater than one.

We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gröbner-Shirshov bases of the Virasoro conformal module and module over the semidirect product of Virasoro conformal algebra and current algebra are given respectively.

The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in Ref. 12, and an increasing sequence DMSH_n of level n, for n ∈ ω, of its subvarieties was investigated in the series Ref. 12–17, of which the present paper is a sequel. It should also be mentioned that the variety DMSH is an equivalent algebraic semantics for the propositional logic, called “De Morgan semi-Heyting logic” which is recently introduced in Ref. 3. Since the lattice of subvarieties of DMSH is dually isomorphic to the lattice of extensions of the De Morgan semi-Heyting logic, the results of this paper and the earlier papers in this series have logical counterparts in the corresponding (axiomatic) extensions of the De Morgan semi-Heyting logic; and, in particular, of De Morgan Heyting logic.

In this paper, we prove two main results: Firstly, we prove that DMSH₁-algebras of leve 1 satisfy Stone identity, generalizing an earlier result that regular DMSH₁-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety DmsStSH of dually ms, Stone semi-Heyting algebras is at level 2. As an application, it is derived that the variety of De Morgan semi-Heyting algebras is also at level 2. It is also shown that these results are sharp.

As a generalization of inverse semigroups, restriction semigroups have arisen in a number of contexts, one being by using relations ${\tilde{\mathcal{R}}}_E$ and ${\tilde{\mathcal{L}}}_E$ which may be regard as the generalized Green relations, the other being as varieties of (bi)-unary semigroups. The purpose of this article is to give a survey of certain techniques used in the study of restriction semigroups. The main objects that we introduce and discuss in this article are W-products, Bruck–Reilly extensions and Szendrei expansions.

The following section is included:

• Author index