Anatolii Illarionovich Shirshov (1921–1981) was an outstanding Russian mat- maticianwhoseworksessentiallyin?uenced thetheoriesofassociative,Lie,Jordan and alternative rings. Many Shirshov’s students and students of his students had a successful research career in mathematics. AnatoliiShirshovwasbornonthe8thofAugustof1921inthevillageKolyvan near Novosibirsk. Before the II World War he started to study mathematics at Tomsk university but then went to the front to ?ght as a volunteer. In 1946 he continued his study at Voroshilovgrad (now Lugansk) Pedagogical Institute and at the same time taught mathematics at a secondary school. In 1950 Shirshov was accepted as a graduate student at the Moscow State University under the supervision of A. G. Kurosh. In 1953 he has successfully defended his Candidate of Science thesis (analog of a Ph. D. ) “Some problems in the theory of nonassociative rings and algebras” and joined the Department of Higher Algebra at the Moscow State University. In 1958 Shirshov was awarded the Doctor of Science degree for the thesis “On some classes of rings that are nearly associative”. In 1960 Shirshov moved to Novosibirsk (at the invitations of S. L. Sobolev and A. I. Malcev) to become one of the founders of the new mathematical institute of the Academy of Sciences (now Sobolev Institute of Mathematics) and to help the formation of the new Novosibirsk State University. From 1960 to 1973 he was a deputy director of the Institute and till his last days he led the research in the theory of algebras at the Institute.

The book is about (associative, Lie and other) algebras, groups, semigroups presented by generators and defining relations. They play a great role in modern mathematics. It is enough to mention the quantum groups and Hopf algebra theory, the Kac-Moody and Borcherds algebra theory, the braid groups and Hecke algebra theory, the Coxeter groups and semisimple Lie algebra theory, the plactic monoid theory. One of the main problems for such presentations is the problem of normal forms of their elements. Classical examples of such normal forms give the Poincaré-Birkhoff-Witt theorem for universal enveloping algebras and Artin-Markov normal form theorem for braid groups in Burau generators.What is now called Gröbner-Shirshov bases theory is a general approach to the problem. It was created by a Russian mathematician A I Shirshov (1921-1981) for Lie algebras (explicitly) and associative algebras (implicitly) in 1962. A few years later, H Hironaka created a theory of standard bases for topological commutative algebra and B Buchberger initiated this kind of theory for commutative algebras, the Gröbner basis theory. The Shirshov paper was largely unknown outside Russia. The book covers this gap in the modern mathematical literature. Now Gröbner-Shirshov bases method has many applications both for classical algebraic structures (associative, Lie algebra, groups, semigroups) and new structures (dialgebra, pre-Lie algebra, Rota-Baxter algebra, operads). This is a general and powerful method in algebra.

This book contains papers presented at the Second International Conference on Algebra, held in Barnaul in August 1991 in honour of the memory of A. I. Shirshov (1921--1981). Many of the results presented here have not been published elsewhere in the literature. The collection provides a panorama of current research in PI-, associative, Lie, and Jordan algebras and discusses the interrelations of these areas with geometry and physics. Other topics in group theory and homological algebra are also covered.

This volume composed of twenty four research articles which are selected from the keynote speakers and invited lectures presented in the 3rd International Congress in Algebra and Combinatorics (ICAC2017) held on 25-28 August 2017 in Hong Kong and one additional invited article. This congress was specially dedicated to Professor Leonid Bokut on the occasion of his 80th birthday.

Even three decades ago, the words 'combinatorial algebra' contrasting, for in stance, the words 'combinatorial topology,' were not a common designation for some branch of mathematics. The collocation 'combinatorial group theory' seems to ap pear first as the title of the book by A. Karras, W. Magnus, and D. Solitar [182] and, later on, it served as the title of the book by R. C. Lyndon and P. Schupp [247]. Nowadays, specialists do not question the existence of 'combinatorial algebra' as a special algebraic activity. The activity is distinguished not only by its objects of research (that are effectively given to some extent) but also by its methods (ef fective to some extent). To be more exact, we could approximately define the term 'combinatorial algebra' for the purposes of this book, as follows: So we call a part of algebra dealing with groups, semi groups , associative algebras, Lie algebras, and other algebraic systems which are given by generators and defining relations {in the first and particular place, free groups, semigroups, algebras, etc. )j a part in which we study universal constructions, viz. free products, lINN-extensions, etc. j and, finally, a part where specific methods such as the Composition Method (in other words, the Diamond Lemma, see [49]) are applied. Surely, the above explanation is far from covering the full scope of the term (compare the prefaces to the books mentioned above).

The book aims to exemplify the recent developments in operad theory, in universal algebra and related topics in algebraic topology and theoretical physics. The conference has established a better connection between mathematicians working on operads (mainly the French team) and mathematicians working in universal algebra (primarily the Chinese team), and to exchange problems, methods and techniques from these two subject areas.

This volume is an outcome of the International Conference on Algebra in celebration of the 70th birthday of Professor Shum Kar-Ping which was held in Gadjah Mada University on 7?10 October 2010. As a consequence of the wide coverage of his research interest and work, it presents 54 research papers, all original and referred, describing the latest research and development, and addressing a variety of issues and methods in semigroups, groups, rings and modules, lattices and Hopf Algebra. The book also provides five well-written expository survey articles which feature the structure of finite groups by A Ballester-Bolinches, R Esteban-Romero, and Yangming Li; new results of Gr”bner-Shirshov basis by L A Bokut, Yuqun Chen, and K P Shum; polygroups and their properties by B Davvaz; main results on abstract characterizations of algebras of n-place functions obtained in the last 40 years by Wieslaw A Dudek and Valentin S Trokhimenko; Inverse semigroups and their generalizations by X M Ren and K P Shum. Recent work on cones of metrics and combinatorics done by M M Deza et al. is included.