Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the con...

We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with algebraic product structure. These have a close relationship $$A_{\infty }$$ and $$C_{\infty morphisms, which classical homotopical tools for analyzing deformations algebraically maps. look at these two different perspectives understand how infinity-morphisms...

We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle probl...

Naively, given a field k and some n > 0, an affine algebraic set in kn is the zero set of some finite collection of polynomial equations with n variables and coefficients in k. A morphism between two affine algebraic sets is a map defined by a tuple of polynomials. Armed with Hilbert’s Nullstellensatz, it isn’t too hard to make this precise in the case that k is algebraically closed, so we begi...

The Jacobian Conjecture is established : If f1, · · · , fn be elements in a polynomial ring k[X1, · · · , Xn] over a field k of characteristic zero such that det(∂fi/∂Xj) is a nonzero constant, then k[f1, · · · , fn] = k[X1, · · · , Xn]. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given...

1 Extension of a Ring by a Module Let C be a ring and N an ideal of C with N = 0. If C ′ = C/N then the C-module N has a canonical C ′-module structure. In a sense analogous with the notion of extension for modules, the data C,N is an “extension” of the ring C ′. Definition 1. Let C ′ be a ring and N a C ′-module. An extension of C ′ by N is a triple (C, ε, i) consisting of a ring C, a surjecti...

The Jacobian Conjecture can be generalized and is established : Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with T = k. Then T = S. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given b...

The Jacobian Conjecture is the following : If φ ∈ Endk(Ank ) with a field k of characteristic zero is unramified, then φ is an automorphism. In this paper, This conjecture is proved affirmatively in the abstract way instead of treating variables in a polynomial ring. Let k be an algebraically closed field, let Ak = Max(k[X1, . . . , Xn]) be an affine space of dimension n over k and let f : Ak −...

The Jacobian Conjecture can be generalized and is established : Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with T = k . Then T = S. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given ...