RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2008, Volume 47, Number 5, Pages 541–557 (Mi al374)

Dominions of universal algebras and projective properties

A. I. Budkin

Abstract: Let $A$ be a universal algebra and $H$ its subalgebra. The dominion of $H$ in $A$ (in a class $\mathcal M$) is the set of all elements $a\in A$ such that every pair of homomorphisms $f,g\colon A\to M\in\mathcal M$ satisfies the following: if $f$ and $g$ coincide on $H$, then $f(a)=g(a)$. A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras $H$ whose dominions coincide with $H$. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup $H$ is closed in each group $\langle H,a\rangle$ generated by one element modulo $H$.

Keywords: universal algebra, dominion, closed algebra, projective property, nilpotent group.

Full text: PDF file (212 kB)
References: PDF file   HTML file

English version:
Algebra and Logic, 2008, 47:5, 304–313

Bibliographic databases:

UDC: 512.57
Revised: 03.09.2008

Citation: A. I. Budkin, “Dominions of universal algebras and projective properties”, Algebra Logika, 47:5 (2008), 541–557; Algebra and Logic, 47:5 (2008), 304–313

Citation in format AMSBIB
\Bibitem{Bud08} \by A.~I.~Budkin \paper Dominions of universal algebras and projective properties \jour Algebra Logika \yr 2008 \vol 47 \issue 5 \pages 541--557 \mathnet{http://mi.mathnet.ru/al374} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2508316} \zmath{https://zbmath.org/?q=an:1164.08313} \transl \jour Algebra and Logic \yr 2008 \vol 47 \issue 5 \pages 304--313 \crossref{https://doi.org/10.1007/s10469-008-9029-6} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261587200002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-57849120217} 

• http://mi.mathnet.ru/eng/al374
• http://mi.mathnet.ru/eng/al/v47/i5/p541

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. I. Budkin, “Dominions in quasivarieties of metabelian groups”, Siberian Math. J., 51:3 (2010), 396–401
2. Budkin A.I., “O dominione polnoi podgruppy metabelevoi gruppy”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-2, 15–19
3. Budkin A.I., “O dominionakh konechnykh podgrupp”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2011, no. 1-2, 15–18
4. A. I. Budkin, “Dominions in Abelian subgroups of metabelian groups”, Algebra and Logic, 51:5 (2012), 404–414
5. A. I. Budkin, “Absolute closedness of torsion-free Abelian groups in the class of metabelian groups”, Algebra and Logic, 53:1 (2014), 9–16
6. S. A. Shakhova, “Absolutely Closed Groups in the Class of $2$-Step Nilpotent Torsion-Free Groups”, Math. Notes, 97:6 (2015), 946–950
7. A. I. Budkin, “Dominions in solvable groups”, Algebra and Logic, 54:5 (2015), 370–379
8. A. I. Budkin, “On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups”, Siberian Math. J., 58:6 (2017), 971–982
9. Campercholi M., “Dominions and Primitive Positive Functions”, J. Symb. Log., 83:1 (2018), 40–54
10. A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Siberian Math. J., 59:4 (2018), 598–609
•  Number of views: This page: 223 Full text: 56 References: 47 First page: 2