23,134 research outputs found

### Composition-Diamond Lemma for Non-associative Algebras over a Commutative Algebra

We establish the Composition-Diamond lemma for non-associative algebras over
a free commutative algebra. As an application, we prove that every countably
generated non-associative algebra over an arbitrary commutative algebra $K$ can
be embedded into a two-generated non-associative algebra over $K$.Comment: 10 page

### Free integro-differential algebras and Groebner-Shirshov bases

The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.
In this paper we apply the method of Groebner-Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota-Baxter algebra on the free differential algebra on the set modulo the differential Rota-Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond Lemma, we consider the free Rota-Baxter algebra on the truncated free differential algebra. A Composition-Diamond Lemma is proved in this context, and a Groebner-Shirshov basis is found for the corresponding differential Rota-Baxter ideal

### On Bergman's Diamond Lemma for Ring Theory

This expository paper deals with the Diamond Lemma for ring theory, which is
proved in the first section of G.M. Bergman, The Diamond Lemma for Ring Theory,
Advances in Mathematics, 29 (1978), pp. 178--218. No originality of the present
note is claimed on the part of the author, except for some suggestions and
figures. Throughout this paper, I shall mostly use Bergman's expressions in his
paper.Comment: 15 page

### Gr\"obner-Shirshov bases for $L$-algebras

In this paper, we firstly establish Composition-Diamond lemma for
$\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra
as a quotient algebra of a free $\Omega$-algebra, and then the normal form of
the free $L$-algebra is obtained. We secondly establish Composition-Diamond
lemma for $L$-algebras. As applications, we give Gr\"{o}bner-Shirshov bases of
the free dialgebra and the free product of two $L$-algebras, and then we show
four embedding theorems of $L$-algebras: 1) Every countably generated
$L$-algebra can be embedded into a two-generated $L$-algebra. 2) Every
$L$-algebra can be embedded into a simple $L$-algebra. 3) Every countably
generated $L$-algebra over a countable field can be embedded into a simple
two-generated $L$-algebra. 4) Three arbitrary $L$-algebras $A$, $B$, $C$ over a
field $k$ can be embedded into a simple $L$-algebra generated by $B$ and $C$ if
$|k|\leq \dim(B*C)$ and $|A|\leq|B*C|$, where $B*C$ is the free product of $B$
and $C$.Comment: 22 page

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