Lattice-Ordered Groupoids and Their Prime Spectrums

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.