a b s t r a c tIn this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free RotaBaxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.

In this survey we give an exposition of the theory of Gröbner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, -algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.Keywords Gröbner basis · Gröbner-Shirshov basis · Composition-Diamond lemma · Congruence · Normal form · Braid group · Free semigroup · Chinese monoid · Plactic monoid · Associative algebra · Lie algebra · Lyndon-Shirshov basis · Lyndon-Shirshov word · PBW theorem · -algebra · Dialgebra · Semiring · Pre-Lie algebra · Rota-Baxter algebra · Category · ModuleSupported by the NNSF of China (11171118) The set of all associative Lyndon-Shirshov words in X NLSW(X)The set of all non-associative Lyndon-Shirshov words in X PBW theoremThe Poincare-Birkhoff-Witt theorem X * The free monoid generated byThe free commutative monoid generated by X X * *The set of all non-associative words (u) in X gp X |S The group generated by X with defining relations S sgp X |S The semigroup generated by X with defining relations S k A field K A commutative algebra over k with unity k X The free associative algebra over k generated by X k X |S The associative algebra over k with generators X and defining relations S S c A Gröbner-Shirshov completion of S I d(S)The ideal generated by a set S sThe maximal word of a polynomial s with respect to some ordering < Irr(S)The set of all monomials avoiding the subwords for all s ∈ S k [X ] The polynomial algebra over k generated by X Lie(X )The free Lie algebra over k generated by X Lie K (X )The free Lie algebra generated by X over a commutative algebra K

Available online xxxx Communicated by Louis Rowen In a memory of M.-P. Schützenberger MSC: 16S15 13P10 20M05 Keywords: Gröbner-Shirshov basis Normal form Associative algebra Plactic monoid Young tableauWe give two explicit (quadratic) presentations of the plactic monoid in row and column generators correspondingly. Then we give direct independent proofs that these presentations are Gröbner-Shirshov bases of the plactic algebra in deg-lex orderings of generators. From Composition-Diamond lemma for associative algebras it follows that the set of Young tableaux is the Knuth normal form for plactic monoid ([30], see also Ch. 5 in [32]).

In this paper, we define the Gröbner-Shirshov bases for a dialgebra. The composition-diamond lemma for dialgebras is given then. As a result, we obtain a Gröbner-Shirshov basis for the universal enveloping algebra of a Leibniz algebra.Definition 2.1 Let k be a field. A k-linear space D equipped with two bilinear multiplications ⊢ and ⊣ is called a dialgebra, if both ⊢ and ⊣ are associative andwhere (v), (w) are diwords in B of length k, l respectively and k + l = n. Proposition 2.3 ([11]) Let D be a dialgebra and B ⊂ D. Any diword of D in the set B is equal to a diword in B of the formAny bracketing of the right side of (1) gives the same result.Definition 2.4 Let X be a set. A free dialgebra D(X) generated by X over k is defined in a usual way by the following commutative diagram: ✲ ❄ ✠ Abstract: In this paper, we define the Gröbner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gröbner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.

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Checkpointing systems are a convenient way for users to make their programs fault-tolerant by intermittently saving program state to disk and restoring that state following a failure. The main concern with checkpointing is the overhead that it adds to running time of the program. This paper describes memory exclusion, an important class of optimizations that reduce the overhead of checkpointing. Some forms of memory exclusion are well-known in the checkpointing community. Others are relatively new. In this paper, we describe all of them within the same framework. We have implemented these optimization techniques in two checkpointers: libckpt, which works on Unix-based workstations, and CLIP, which works on the Intel Paragon. Both checkpointers are publicly available at no cost. We have checkpointed various long-running applications with both checkpointers and have explored the performance improvements that may be gained through memory exclusion. Results from these experiments are presented and show the improvements in time and space overhead. Excluding the code segment and using the stack pointerCheckpoints store the address space of a processor. Typically these address spaces have four segments: executable code, global data, heap and stack. In most computer systems, the

In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov Ω-words on a set X and prove that the set of all non-associative Lyndon-Shirshov Ω-words forms a linear basis of the free Lie Ω-algebra on the set X. From this, we establish Gröbner-Shirshov bases theory for Lie Ω-algebras. As applications, we give Gröbner-Shirshov bases for free λ-Rota-Baxter Lie algebras, free modified λ-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.

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