PBW-Pairs of Varieties of Linear Algebras

The notion of a Poincaré–Birkhoff–Witt (PBW)-pair of varieties of linear algebras over a field is under consideration. Examples of PBW-pairs are given. We prove that if (𝒱, 𝒲) is a PBW-pair and the variety 𝒱 is homogeneous and Schreier, then so is 𝒲; the results similar to the Schreier property for PBW-pairs are also true for the Freiheitssatz and Word problem. In particular, it follows that the Freiheitssatz is true for the varieties of Akivis and Sabinin algebras. We give also examples of varieties that do not satisfy the Freiheitssatz. It is shown that an element u of a free algebra 𝒲[X] in a homogeneous Schreier variety of algebras 𝒲 satisfying the Freiheitssatz is a primitive element (a coordinate polynomial) if and only if the factor algebra of 𝒲[X] by the ideal generated by the element u is a free algebra in 𝒲. We consider also properties of primitive elements.     

When Summands of Completely Decomposable Modules Are Completely Decomposable

We examine when summands of completely decomposable modules over a domain R are again completely decomposable. We show that this is the case if R is an h-local Prüfer domain. If R is 1-dimensional Noetherian, then the problem reduces locally if almost all localizations are integrally closed. If R is 1-dimensional Noetherian and local, then the integral closure of R must have at most two maximal ideals.     

The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.     

Study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge

A study of the zero modes of the Faddeev–Popov operator in the maximal Abelian gauge is presented in the case of the gauge group SU(2)$SU\left(2\right)$ and for different Euclidean space–time dimensions. Explicit examples of classes of normalizable zero modes and corresponding gauge field configurations are constructed by taking into account two boundary conditions, namely: (i) the finite Euclidean Yang–Mills action, (ii) the finite Hilbert norm.     

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

Errata to ``Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties'

We make a correction to Remark 4.3 and the proof of Theorem 4.2 (Peterson's Theorem) which identifies with the coordinate ring of a certain affine stratum of the Peterson variety . Explicitly, we introduce additional coordinates to obtain a complete coordinate system on and then show that they lie in the defining ideal of the Peterson variety , hence play no role in the presentation of .

     

Fixed points avoiding Abelian k-powers

We show that the problem of whether the fixed point of a morphism avoids Abelian k-powers is decidable under rather general conditions, the most important being that the frequency matrix M of the morphism be invertible and that |M−1|<1, where |⋅| denotes a certain matrix norm.     

Index sets of computable structures

The index set of a computable structure is the set of indices for computable copies of . We determine complexity of the index sets of various mathematically interesting structures including different finite structures, ℚ-vector spaces, Archimedean real-closed ordered fields, reduced Abelian p-groups of length less than ω², and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-complete Π _n ⁰ , d-Σ _n ⁰ , or Σ _n ⁰ , for various n. In each case the calculation involves finding an optimal sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Π_n, d-Σ_n, or Σ_n) yields a bound on the complexity of the index set. Whenever we show m-completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey’s theory. Supported by the NSF grants DMS-0139626 and DMS-0353748. Supported by the NSF grant DMS-0502499 and by the Columbian Research Fellowship of the George Washington University. Supported by the NSF grant DMS-0353748. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 538–574, September–October, 2006.