Let
F be an infinite field of characteristic
p
>
2
and let
E be the Grassmann algebra generated by an infinite dimensional vector space
V over
F. In this paper, we describe the
T
2
-ideal of the
Z
2
...-graded polynomial identities of the Grassmann algebra
E for any
Z
2
-grading such that
V is homogeneous in the grading. In particular, we give a description of the
T
2
-ideal of the graded identities of
E in the case there is a finite number of homogeneous elements of the linear basis of
E belonging to one of the homogenous components of
E
.
Let
be a field of characteristic 0 and L be a G-graded Lie PI-algebra where the support of L is a finite subset of G. We define the G-graded Gelfand-Kirillov dimension (GK) of L in k-variables as the ...GK dimension of its G-graded relatively free algebra having k homogeneous variables for each element of the support of L. We compute the G-graded GK dimension of
, where G is any abelian group. Then, we compute the exact value for the
-graded GK dimension of
endowed with the
-grading of Vasilovsky.
Let F be a field of characteristic 0. We consider the upper triangular matrices with entries in F of size 2, 3 and 4 endowed with the grading induced by that of Vasilovsky. In this paper we give ...explicit computation for the multiplicities of the Y-proper graded cocharacters and codimensions of these algebras.
By considering E the infinite dimensional Grassmann algebra over a field of characteristic zero and the T-prime algebra M = M2(E) with the Z2-grading ...{/ampp/image?path=/713597239/923055307/lagb_a_405694_o_ilm0001.gif } , in this article we describe the space of the graded central polynomials modulo the ideal of the graded identities of M.
Let A, B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let IdG(A) be the set of graded identities of A (resp. IdG(B)). ...We show that if A, B are G-simple then there is a graded embedding ϕ:A→B if and only if IdG(B)⊆IdG(A). We also give a weaker generalization for the case where A is G-semisimple and B is arbitrary.
Let
G
be an arbitrary abelian group and let
A
and
B
be two finite dimensional
G
-graded simple algebras over an algebraically closed field
F
such that the orders of all finite subgroups of
G
are ...invertible in
F
. We prove that
A
and
B
are isomorphic if and only if they satisfy the same
G
-graded identities. We also describe all isomorphism classes of finite dimensional
G
-graded simple algebras.
We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a ...G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ℤ
2
)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.
Let
K be an infinite field and let
UT
n
(
K) denote the algebra of
n×
n upper triangular matrices over
K. We describe all elementary gradings on this algebra. Further we describe the generators of ...the ideals of graded polynomial identities of
UT
n
(
K) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group.
The paper considers an early approach toward a (fuzzy) set theory with a graded membership predicate and a graded equality relation which had been developed by the German mathematician Klaua in 1965.
...In the context of the mathematical fuzzy logic
MTL of left-continuous t-norms we discuss some properties of these graded relations. We compare the simultaneous recursive definitions of these relations with the very similar approach toward Boolean algebra valued interpretations of membership and equality, presented in 1967 by Scott and R. Solovay in the context of independence proofs for
ZF set theory.
Finally we speculate about possible reasons why Klaua soon abandoned this approach.
The Jordan algebra of the symmetric matrices of order two over a field K has two natural gradings by Z2, the cyclic group of order 2. We describe the graded polynomial identities for these two ...gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a Z2-grading on Bn, the Jordan algebra of a symmetric bilinear form in a vector space of dimension n (n=1,2,…,∞). We call this grading the scalar one since its even part consists only of the scalars. As a by-product we obtain finite bases of the Z2-graded identities for Bn. In fact the last result describes the weak Jordan polynomial identities for the pair (Bn,Vn).
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