New Trends in Algebras and Combinatorics - Proceedings of the Third International Congress in Algebras and Combinatorics (ICAC2017)

Contents
Preface
A short biography of Professor Leonid Bokut
My life in mathematics, 60 years
1. My education and positions
2. My teachers
3. Main results
4. My activities in mathematical life
5. List of my students
6. Main publications
6.1. Books
6.2. Papers
Acknowledgments
The incipience of Gröobner–Shirshov bases
Some new results on cluttered ordering on several special bipartite graphs
1. Introduction
2. Known Results
3. The initial stages for a bipartite graph H(1; k; t)
4. A Cluttered Ordering of H(1; k; 2t)
Acknowledgement
References
On n-ary polynomially complete quasigroups
1. Introduction
2. Isotopies of n quasigroups
3. Affine n-quasgroups
4. Polynomial completeness
5. Construction of ternary polynomially complete quasigroups
References
Matrix rings as one sided σ-(S, 1) rings
1. Introduction and Preliminaries
2. σ-(S, 1) rings
3. One sided σ – (S, 1) rings
References
Embedding of post-Lie algebras into postassociative algebras
1. Introduction
2. Postalgebras
3. Embedding of pre- and postalgebras into RB-algebras
4. Embedding of post-Lie algebras into postassociative algebras
Acknowledgements
References
Free left GC-lpp semigroups
1. Introduction
2. Quasi-variety of left GC-lpp semigroups
3. Free left GC-lpp semigroups
4. Semidirect product and free left GC-lpp semigroups
5. A proper cover theorem for left GC-lpp semigroups
Acknowledgement
References
On the cover-avoiding properties in finite groups
1. Introduction
2. Cover-avoiding properties and the structure of groups
3. Semi cover-avoiding properties and the structure of groups
4. Further investigations
References
The length of the group algebra of the group Q8
1. Introduction
2. Some known facts
3. Length of the algebra M2(F) Dn(F)
4. On generators of diagonal matrices
5. Generalized quaternion algebras
6. Some technical results
7. Length of D D4(F)
8. Length of the quaternion group algebra
9. Fields of characteristics 2
References
Results on intersecting families of subsets, a survey
1. Introduction
2. t-intersecting families
3. Largest union-intersecting families and related problems
4. Two- or more-part intersecting families
5. Minimum shadows of t-intersecting families
References
Powers of monomial ideals and combinatorics
1. Introduction
2. Preliminaries
3. Stability of associated primes
3.1. Associated primes of integral closures of powers
3.2. Associated primes of powers
3.3. Case of edge ideals and square-free monomial ideals
4. Stability of Depth
4.1. Depth of powers of integral closures
4.2. Depth of symbolic powers
4.3. Depth of powers
4.4. Cohen-Macaulay property of powers
Acknowledgment
References
Quantum calculus. An introduction
1. Introduction
2. q-Analogs of Numbers
2.1. q-Analogs of non-negative Integers
2.2. q-Analogs of Factorial
2.3. q-Analogs of complex numbers
3. q-Pochhammer Symbol
4. q-Analogs and Vector spaces
4.1. q-Factorial and Vector Spaces
4.2. q-Binomial Coefficients and Vector Spaces
4.3. q-Analog of the Symmetric Groups
5. q-Weyl Calculus
5.1. What are q-Weyl pairs?
6. q-Binomial Theorem
6.1. q-Binomial Coefficients or Gaussian Polynomials
6.2. q-analog of the Binomial Theorem
7. q-Derivative or Jackson's Derivative
8. q-Binomial Theorem
9. q-Trigonometry
9.1. q-Exponentials
9.2. q-cosine and q-sine functions
10. Jackson's Integral
11. Jacobi Elliptic Functions & Theta Functions
11.1. Elliptic Functions
11.2. Theta Functions
References
Tree breadth of the continued fractions root Finding method
1. Introduction
2. Definitions
2.1. Effect of Möbius Transforms
2.2. Bounds on the number of roots
3. Bound on the tree breadth for a polynomial
4. Bound on the expected breadth of the transformation tree
4.1. Expected breadth
4.2. Alternative hypothesis
4.3. Dependence of expected breadth on the real roots
4.4. Experimental evidence
Acknowledgments
References
Combinatorial rank of quantum groups of infinite series
1. Introduction
2. Preliminaries
2.1. Skew brackets
2.2. Radford biproduct and the ideal Λ
2.3. Differential calculi
3. Combinatorial representation
4. Hard super-letters and PBW basis
5. PBW generators for u+q (sp2n)
6. Defining relations for G(X)/J
7. Constants of differential calculi
8. Combinatorial rank
Acknowledgments
References
Gröbner–Shirshov bases for associative conformal algebras with arbitrary locality function
1. Introduction
2. Preliminaries: CD-Lemma for modules
3. Conformal algebras
4. Module construction of the free associative conformal algebra
5. Applications
References
The groups Gkk+1 and fundamental groups of configuration spaces
1. Introduction
2. Basic definitions
3. The realisability of Gkk+1
3.1. Constructing a braid from a word in Gkk+1
3.1. Constructing a braid from a word in Gkk+1
4. The group Hk and the algebraic lemma
5. Concluding remarks
References
Presentations of inverse semigroups: progress, problems and applications
1. Introduction
2. Preliminaries
3. Some applications of Stephen's procedure
4. One-relator inverse monoids
5. Adian inverse semigroups
6. Inverse semigroups and operator algebras
7. Inverse monoids and immersions of cell complexes
References
Replicators, Manin white product of binary operads and average operators
1. Introduction
2. The replicators of a binary operad
2.1. Replicators of planar binary trees
2.1.1. Labeled trees
2.1.2. Duplicators
2.1.3. Triplicators
2.2. Replicators of binary operads
3. Examples of duplicators and triplicators
3.1. The nonsymmetric case
3.2. The symmetric case
3.2.1. Examples of duplicators
3.2.2. Examples of triplicators
4. Operads, their duplicators and triplicators
4.1. Relationship between an operad and its duplicator and triplicator
4.2. Relationship between the duplicator and the triplicator of a binary operad
5. Duality of replicators with successors and Manin products
5.1. The duality of replicators with successors
5.2. Replicators and Manin white products
6. Replicators and Average operators on operads
6.1. Duplicators and di-average operators
6.2. Triplicators and tri-average operators
Acknowledgments
References
Kac-Moody Groups and Their Representations
1. Introduction
2. Representations of uncompleted group
2.1. Kac-Moody Lie algebra
2.2. Kac-Moody group
2.3. Adjoint representation
2.4. Over-restricted representations
3. Representations of completed group
3.1. Completion
3.2. Comparison to other completions
3.3. Davis Building
3.4. Projective Dimension of Smooth Representations
3.5. Localisation
4. Questions
4.1. Isomorphism Problem
4.2. Existence of Restricted Structure
4.3. Humphreys-Verma Conjecture
4.4. Theory of Over-restricted Representations
4.5. Congruence Kernel
4.6. Lattices in Locally Pro-p-complete Kac-Moody Groups
4.7. Completions
4.8. Schneider-Stuhler Resolution
4.9. Homology of CAT(0)-Complex
References
On the heredity of V-modules over Noetherian nonsingular rings
1. Introduction
2. Semiprime Noetherian Rings with nonsingular V-modules
3. For Non-semiprime Case, When Is A UR(P)-projective V-Module Hereditary?
References
Quasigroups, hyperquasigroups, and vector spaces over fields with one or more elements
1. Introduction
2. Quasigroups, and fields with one or more elements
2.1. Combinatorial quasigroups
2.2. Equational quasigroups
2.3. Cayley's Theorem
2.4. Fields with one or more elements
3. Hyperquasigroups and linear combinations
3.1. Hyperquasigroups
3.2. Linear hyperquasigroups
3.3. Linear combinations
3.4. Spans and subspaces
3.5. Linear transformations
4. Categories of relations
4.1. Relations between sets
4.2. Linear algebra
4.3. Counting
5. Conclusion
References
Gröbner-Shirshov bases for associative conformal modules
1. Introduction
2. Preliminaries
2.1. Associative (Lie) conformal algebras
2.2. Modules over a Lie or associative conformal algebra
3. Free associative conformal modules
3.1. Double free associative conformal modules
3.2. Free associative conformal C-modules
4. Composition-Diamond lemma for associative conformal modules
4.1. A monomial ordering
4.2. S-words and normal S-words
4.3. Compositions
4.4. Key lemmas
4.5. Composition-Diamond lemma for associative conformal modules
5. Applications
5.1. Conformal modules over universal enveloping conformal algebra
Acknowledgement
References
De Morgan semi-Heyting and Heyting algebras
1. Introduction
2. Preliminaries
3. The variety DM1 of De Morgan semi-Heyting algebras of level 1
4. The level of the variety DmsSt of dually ms, Stone semi-Heyting algebras
References
Restriction semigroups
Introduction
1. Preliminaries
1.1. Restriction semigroups
1.2. Morphisms and congruences
1.3. Various proper restriction semigroups
2. W-products
2.1. Left restriction semigroups
2.2. Free restriction semigroups
2.3. F-restriction semigroups.
2.4. (Almost) factorizable restriction semigroups
3. Generalized W-product
4. The construction ST;R
5. Bruck–Reilly extensions
6. Szendrei expansions
References
Author index