Congruences on bicyclic extensions of a linearly ordered group

In the paper we study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and \bar{\mathscr{B}}\,^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. We describe Green's relations on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and \bar{\mathscr{B}}\,^+(G), their bands and show that they are simple, and moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences on the semigroup $\mathscr{B}(G)$ (and $\mathscr{B}^+(G)$) are group congruences if and only if the group $G$ is archimedean. Also we describe the structure of group congruences on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and \bar{\mathscr{B}}\,^+(G).Comment: 20 page

On the geometric quantization of Jacobi manifolds

29 pages.The geometric quantization of Jacobi manifolds is discussed. A natural cohomology (termed Lichnerowicz–Jacobi) on a Jacobi manifold is introduced, and using it the existence of prequantization bundles is characterized. To do this, a notion of contravariant derivatives is used, in such a way that the procedure developed by Vaisman for Poisson manifolds is naturally extended. A notion of polarization is discussed and the quantization problem is studied. The existence of prequantization representations is also considered.This work has been partially supported through grants DGICYT (Spain) (Project No. PB94- 0106) and University of La Laguna (Spain).Peer reviewe

On chains in HH-closed topological pospaces

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.Comment: We have rewritten and substantially expanded the manuscrip

O verigah v H-zaprtih topoloških delno urejenih prostorih

Preučujemo verige v ▫$H$▫-zaprtem topološkem delno urejenem prostoru. Podamo zadostne pogoje za maksimalno verigo ▫$L$▫ v ▫$H$▫-zaprtem topološkem delno urejenem prstoru (▫$H$▫-zaprti topološki polmreži), pri katerih ▫$L$▫ vsebuje maksimalen (minimalen) element. Podamo tudi zadostne pogoje, pri katerih je linearno urejen topološki delno urejen prostor ▫$H$▫-zaprt. Dokažemo, da je linearno urejena ▫$H$▫-zaprta topološka polmreža ▫$H$▫-zaprt topološki delno urejen prostor in, da to v splošnem ne velja. Konstruiramo primer ▫$H$▫-zaprtega topološkega delno urejenega prostora z nezaprto ▫$H$▫-zaprto maksimalno verigo ter podamo zadostne pogoje, pri katerih je maksimalna veriga ▫$H$▫-zaprtega topološkega delno urejenega prostora ▫$H$▫-zaprt topološki delno urejen prostor.We study chains in an ▫$H$▫-closed topological partially ordered space. We give sufficient conditions for a maximal chain ▫$L$▫ in an ▫$H$▫-closed topological partially ordered space (▫$H$▫-closed topological semilattice) under which ▫$L$▫ contains a maximal (minimal) element. We also give sufficient conditions for a linearly ordered topological partially ordered space to be ▫$H$▫-closed. We prove that a linearly ordered ▫$H$▫-closed topological semilattice is an ▫$H$▫-closed topological pospace and show that in general, this is not true. We construct an example of an ▫$H$▫-closed topological pospace with a non-▫$H$▫-closed maximal chain and give sufficient conditions under which a maximal chain of an ▫$H$▫-closed topological pospace is an ▫$H$▫-closed topological pospac

O kodimenzijski rasti enostavnih barvnih Liejevih algeber

Predmet naših raziskav so polinomske identitete končno razsežnih enostavnih barvnih Liejevih superalgeber nad algebrsko zaprtim poljem z ničelno karakteristiko, gradacijo katerih podaja produkt dveh cikličnih grup reda 2. Dokazujemo, da kodimenzije opisanih identitet naraščajo eksponentno, stopnja te rasti pa je enaka razsežnosti dane algebre. Podoben rezultat smo dobili tudi za gradirane identitete in gradirane kodimenzije.We study polynomial identities of finite dimensional simple color Lie superalgebras over an algebraically closed field of characteristic zero graded by the product of two cyclic groups of order 2. We prove that the codimensions of identities grow exponentially and the rate of exponent equals the dimension of the algebra. A similar result is also obtained for graded identities and graded codimensions

On Jordan ∗-mappings in rings with involution

The objective of this paper is to study Jordan ∗-mappings in rings with involution ∗. In particular, we prove that if R is a prime ring with involution ∗, of characteristic different from 2 and D is a nonzero Jordan ∗-derivation of R such that [D(x),x]=0, for all x∈R and S(R)∩Z(R)≠(0), then R is commutative. Further, we also prove a similar result in the setting of Jordan left ∗-derivation. Finally, we prove that any symmetric Jordan triple ∗-biderivation on a 2-torsion free semiprime ring with involution ∗ is a symmetric Jordan ∗-biderivation