c-Regular cyclically ordered groups

We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex numbers, that every discrete c-regular cyclically ordered abelian group is elementarily equivalent to some ultraproduct of finite cyclic groups, and that the discrete regular non-c-regular cyclically ordered abelian groups are elementarily equivalent to the linearly cyclically ordered group of integers.Comment: 20 page

Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group W is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied in Boothby et al.. In particular the authors enumerated cyclically fully commutative elements in all Coxeter groups having a finite number of them. In this work we characterize and enumerate cyclically fully commutative elements according to their Coxeter length in all finite or affine Coxeter groups by using a new operation on heaps, the cylindric transformation. In finite types, this refines the work of Boothby et al., by adding a new parameter. In affine type, all the results are new. In particular, we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine Coxeter groups. We study afterwards the cyclically fully commutative involutions and prove that their number is finite in all Coxeter groups.Comment: 23 pages, 16 figure

Some criteria of cyclically pure injective modules

The structure of cyclically pure injective modules over a commutative ring $R$ is investigated and several characterizations for them are presented. In particular, we prove that a module $D$ is cyclically pure injective if and only if $D$ is isomorphic to a direct summand of a module of the form \Hom_R(L,E) where $L$ is the direct sum of a family of finitely presented cyclic modules and $E$ is an injective module. Also, we prove that over a quasi-complete Noetherian ring (R,\fm) an $R$-module $D$ is cyclically pure injective if and only if there is a family $\{C_\lambda\}_{\lambda\in \Lambda}$ of cocyclic modules such that $D$ is isomorphic to a direct summand of $\Pi_{\lambda\in \Lambda}C_\lambda$. Finally, we show that over a complete local ring every finitely generated module which has small cofinite irreducibles is cyclically pure injective.Comment: 18 pages, to appear in Journal of Algebr

Cyclically presented modules, projective covers and factorizations

We investigate projective covers of cyclically presented modules, characterizing the rings over which every cyclically presented module has a projective cover as the rings $R$ that are Von Neumann regular modulo their Jacobson radical $J(R)$ and in which idempotents can be lifted modulo $J(R)$. Cyclically presented modules naturally appear in the study of factorizations of elements in non-necessarily commutative integral domains. One of the possible applications is to the modules $M_R$ whose endomorphism ring $E:=(M_R)$ is Von Neumann regular modulo $J(E)$ and in which idempotents lift modulo $J(E)$.Comment: 17 page

Palindromic primitives and palindromic bases in the free group of rank two

The present paper records more details of the relationship between primitive elements and palindromes in F_2, the free group of rank two. We characterise the conjugacy classes of primitive elements which contain palindromes as those which contain cyclically reduced words of odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F_2. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.Comment: 8 page

On Cyclically Symmetrical Spacetimes

In a recent paper Carot et al. considered the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this paper a similar result, valid under considerably weaker assumptions, is derived: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I, II, III, VII_0, VIII or IX. Some results on n-dimensional Lie groups are also derived and applied to show there are severe restrictions on the structure of the allowed four-dimensional Lie transformation groups compatible with cyclic symmetry.Comment: 6 pages, LaTex. (World Scientific style file: sprocl.sty needed) To appear in Proceedings of the Spanish Relativity Meeting (EREs2000), World Scientific Publishin