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Lattice-Ordered Groupoids and Their Prime Spectrums

In: Algebra and Operator Theory

Author

Listed:
  • Dj. Khadjiev

    (Tashkent State University, Departement of Mathematics)

  • T. M. Shamilev

    (Simferopol Engineering Pedagogical Institute)

Abstract

Our main results are the following: 1. Let L be a complete ordered groupoid ([1], ch. XIV). We introduce definitions of r -radical and R -radical elements in L and describe some their properties. 2. Let L be a complete ordered groupoid in which every element is ideal. Denote by L r the lattice of all r-radical elements in L. Then L r satisfies the infinite ∧-distributive condition: $$ a \wedge \left( {{{ \vee }_{{\tau \in T}}}{{b}_{\tau }}} \right) = {{ \vee }_{{\tau \in T}}}\left( {a \wedge {{b}_{\tau }}} \right) $$ for any a,b r ∈ L r ,r ∈ T. Let L be a complete ordered groupoid in which every element is ideal and r-radical. Then a b=a^b for any a,b ∈ L and L satisfies the infinite ^- distributive condition. Analogous statements are hold for R –radical elements.

Suggested Citation

  • Dj. Khadjiev & T. M. Shamilev, 1998. "Lattice-Ordered Groupoids and Their Prime Spectrums," Springer Books, in: Yusupdjan Khakimdjanov & Michel Goze & Shavkat A. Ayupov (ed.), Algebra and Operator Theory, pages 157-164, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-5072-9_13
    DOI: 10.1007/978-94-011-5072-9_13
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