Let $X$ be a $BCH$-algebra and $I$ be an ideal of $X$. In this paper, we introduce the concept of $sqrt{I}$. We show that it is an ideal of $X$, when $I$ is closed ideal of $X$. Then we verify some useful properties of it. We prove that it is the ::::union:::: of all $k-$nil ideals of $I$. Moreover, if $I$ is a closed ideal of $X$, then $sqrt{I}$ is a closed translation ideal and so we can cons...

The algebra Mn(K) of the matrices n×n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z), where α is the degree of the corresponding variable. This is a generalization of a result of...

Let A1 = K〈X,Y | [Y,X] = 1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(Y X) of A1 is Z. Let A1 → A1, X 7→ x, Y 7→ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of theWeyl algebra A1 is Z and the eigenvector algebra of ad(yx) is K〈x...

K-extended basic macro grammars are introduced, where K is any class of languages. The class B(K) of languages generated by such grammars is investigated, together with the class LB(K) of languages generated by the corresponding linear basic grammars. For any full semiAFL K, B(K) is a full AFL closed under iterated LB(K)-substitution, but not necessarily under substitution. For any machine type...

Let R be monomial sub-algebra of k[x1, . . . , xN ] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x1, . . . , xN and whose edges are {(xi, xj)|xixj ∈ R}. Conversely, for any graph G with vertices {x1, . . . , xN} we define the edge algebra associ...

The algebra Sn of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting, left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra Sn is a noncommutative, non-Noetherian algebra of classical Krull dimension 2n and of global dimension n and is not a domain. If the ground field K has characteristic zero then the alg...

In this note first we define the notions of fuzzy positive implicativehyper BCK-ideals of types 1,2,3 and 4. Then we prove some theorems whichcharacterize the above notions according to the level subsets. Also we obtainthe relationships among these notions, fuzzy (strong, weak, reflexive) hyperBCK-ideals and fuzzy positive implicative hyper BCK-ideals of types 5,6,7and 8. Then, we define the no...

We present a hyperproduct on non square matrices by using a generalization of the well known P-hopes. This theory is connected with the corresponding classical algebra, mainly with the theory of representations by (hyper) matrices. This can be achieved by using the fundamental relations defined on the hyperstructures.

in this paper we consider c0-group of unitary operators on a hilbert c*-module e. in particular we show that if a?l(e) be a c*-algebra including k(e) and ?t a c0-group of *-automorphisms on a, such that there is x?e with =1 and ?t (?x,x) = ?x,x t?r, then there is a c0-group ut of unitaries in l(e) such that ?t(a) = ut a ut*.