Generalized commutativity of lattice-ordered groups II

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• 作者：M. R. Darnel ; W. C. Holland ; H. Pajoohesh
• 关键词：Primary ; 06F15 ; Secondary ; 08B15 ; 20F60 ; 20K99 ; lattice ; ordered group ; variety ; commutativity
• 刊名：Algebra Universalis
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：75
• 期：1
• 页码：51-59
• 全文大小：575 KB
• 参考文献：1.Darnel, M.R.: Theory of Lattice-Ordered Groups. Marcel Dekker (1995)
  2.Darnel M.R., Holland W.C.: Minimal non-metabelian varieties of \({\ell}\) -groups that contain no nonabelian o-groups. Comm. Alg. 42, 5100–5133 (2014)MATH MathSciNet CrossRef
  3.Darnel M.R., Holland W.C., Pajoohesh H.: Generalized commutativity of lattice-ordered groups. Math. Slovaca 65, 325–342 (2015)MathSciNet
  4.Holland W.C., Mekler A.H., Reilly N.R.: Varieties of lattice-ordered groups in which prime powers commute. Algebra Universalis 23, 196–214 (1986)MATH MathSciNet CrossRef
  5.Botto Mura, R., Rhemtulla, A.: Orderable Groups. Lecture Notes in Pure and Appl. Math. 27, Marcel Dekker, New York (1977)
  6.Reilly, N.R., Varieties of lattice-ordered groups. In: Glass, A.M.W., Holland, W.C. (eds.) Lattice-Ordered Groups pp. 228–277. Kluwer, Dordrecht (1989)
  
• 作者单位：M. R. Darnel (1)
  W. C. Holland (2)
  H. Pajoohesh (3)
  
  1. Indiana University South Bend, South Bend, USA
  2. University of Colorado, Boulder, USA
  3. Department of Mathematics, Medgar Evers College, CUNY, Brooklyn, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8911

文摘

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n. We will now generalize this to consider an \({\ell}\)-group G in which for two fixed integers \({m, n, x^{m}y^{n} = y^{n}x^{m}}\) for all \({x, y \in G}\). Then we will generalize this to a set of more than two integers. Finally, we will consider an even more general situation where one or both of the exponents are not fixed. Key words and phrases lattice-ordered group variety commutativity