Not Distributive Lattice over an L1 (L)– Space

Abstract

1. Lindenstrauss & H. P. Rosenthal in ^[1] The have claimed that there are sufficiently many Boolean algebras of projections. ‘Classification of spatial ^[2] by
   N. Christopher Phillips and Maria Grazia Viola have discussed the specific class of operator algebras. The duo again discussed Cuntz algebras in ^[3] on a commutative Euclidean ring in which every non zero element possess multiplicative inverse forming a field . The n+1 copies of    whose direct sum and scalar product operations can help to construct a polynomial ring having polynomials of degree less than or equal to n that also can be verified as the metric space by suitably constructing the distance function d as the metric. Sergio A. Alwarez ^[4] has given arithmetic on  that promoted the Minkowski inequality including the definitions in the introduction chapter followed. Also it can be verified as a linear space of dimension n +1. A metric space upon which the function norm is introduced and taking the collection of convergent sequences over it forms a ^[2] Banach algebra /Banach space or ^[1] an L^p- space. On constructing bases for ideals in polynomial rings over the integers ^[6] , Ayoub C.W. claimed that  Suitably defining the partial order, join and meet operations upon it, the smallest - algebra containing  is  will be a lattice. Using the inequality of the sum of the determinants greater than the determinant of the sum, the distributive property is shown to be failed in this lattice.