Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.     

Compatibility support mappings in effect algebras

We give a characterization of subsets of effect algebras, that can be embedded into a range of an observable. To give this characterization, we introduce a new notion of compatibility support mappings.     

Ternary mappings of triangular algebras

Aequationes mathematicae - We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of...     

k-commuting mappings of generalized matrix algebras

Periodica Mathematica Hungarica - In this paper we will study k-commuting mappings of generalized matrix algebras. The general form of arbitrary k-commuting mapping of a generalized matrix algebra...     

The isotone mappings on Ockham algebras

In this paper we shall describe some algebraic concepts of Ockham algebras. We show that, ifL, M K _1,1 thenH(S(L),S(M)) S(H(L,M)).     

Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A ^(m) be the m-replicated algebra of A. We prove that the representation dimension of A ^(m) is at most 3, and that the dominant dimension of A ^(m) is at least m.     

Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If \(M\in \mathcal {C}({\Phi })\), then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P?∧ ^t P⊕R[M] ^t?1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P?∧ ^t P⊕R[M] ^t?1.     

Representation dimension of exterior algebras

We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F₂ of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.     

Quantization dimension and temperature function for bi-Lipschitz mappings

The quantization dimension function for a probability measure supported on the limit set generated by a set of contractive bi-Lipschitz mappings is determined. The relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established.     

Minimal algebras and 2-step nilpotent Lie algebras in dimension 7

We use the methods of Bazzoni and Muñoz (Trans Am Math Soc 364:1007–1028, 2012) to give a classification of 7-dimensional minimal algebras, generated in degree 1, over any field ${\mathbf{k}}$ of characteristic ${{\rm char}(\mathbf{k})\neq 2}$ , whose characteristic filtration has length 2. Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.     

Krull dimension and integrality of symmetric algebras

For a finitely generated moduleE over the Noetherian ringR we consider formulas for the Krull dimension of the symmetric algebraS(E). A result of Huneke and Rossi is re-proved and an effective formula is derived that reads the dimension ofS(E) from a presentation ofE. They provide a first line of obstructions forS(E) to be an integral domain. For algebras of codimension at most four we give methods, including computer-assisted ones, to ascertain whetherS(E) is a Cohen-Macaulay domain.AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 13H10; Secondary 13-04, 13C05, 13C15Partially supported by CNPq, BrazilPartially supported by the NSF     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

Affine mappings in the geometries of algebras

Modules and algebras over a commutative ring R and the geometry associated with an algebra are studied. The non-homogeneous members of a module, for which no maximal ideal of R contains every co-ordinate, the generators of a module, for which polynomials in the co-ordinates take arbitrary values and the non-singular members of an algebra are investigated in relation to the group G of non-singular R-linear transformations. Particular attention is given to bi-quadratic extensions of fields. Geometrical isomorphisms are proved to be exactly those transformations that can be written as the composition of a translation, a member of G and a co-ordinate-wise extension to the algebra of an automorphism of R.The author gratefully acknowledges the help and advice of Prof. Walter Benz of the University of Waterloo and the financial support of the University of Waterloo and the National Research Council of Canada.     

On the dimension of graded algebras

To each graded algebra R with a finite number of generators we associate the series T(R, z) = d_nzⁿ, where d_n is the dimension of the homogeneous component of R. It is proved that if the dimensions d_n have polynomial growth, then the Krull dimension of R cannot exceed the order of the pole of the series T(R, z) for z=1 by more than 1.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 209–216, August, 1973.     

Krull dimension for differential graded algebras

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring and compare it to the system of parameters defined by Christensen. We show that these notions differ in general but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals.     

Finitistic dimension of standardly stratified algebras

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.