Representation of ABA’s by Sections of Sheaves

Chudamani, R.; Prasad, K. Rama; Rao, K. Krishna; Swamy, U.M.

doi:10.28924/2291-8639-21-2023-105

Research Article

Representation of ABA’s by Sections of Sheaves

R. Chudamani
K. Rama Prasad*
K. Krishna Rao
U.M. Swamy

* Corresponding author

Abstract

An Almost Boolean Algebra (A, ∧, ∨, 0) (abbreviated as ABA) is an Almost Distributive Lattice (ADL) with a maximal element in which for any x∈A, there exists y∈A such that x∧y = 0 and x∨y is a maximal element in A. If (S, Π, X) is a sheaf of nontrivial discrete ADL’s over a Boolean space such that for any global section f, support of f is open, then it is proved that the set Γ(X, S) of all global sections is an ABA. Conversely, it is proved that every ABA is isomorphic to the ADL of global sections of a suitable sheaf of discrete ADL’s over a Boolean space.

How to Cite

R. Chudamani, K. R. Prasad, K. K. Rao, U. M. Swamy, Representation of ABA’s by Sections of Sheaves, Int. J. Anal. Appl. 21 (2023), 105. https://doi.org/10.28924/2291-8639-21-2023-105

BibTeX

@article{Chudamani2023,
  author  = {Chudamani, R. and Prasad, K. Rama and Rao, K. Krishna and Swamy, U.M.},
  title   = {Representation of ABA’s by Sections of Sheaves},
  journal = {Int. J. Anal. Appl.},
  volume  = {21},
  year    = {2023},
  pages   = {105},
  doi     = {10.28924/2291-8639-21-2023-105}
}

References

[1] G. Grätzer, Universal Algebra, Van Nostrand, Princeton, N. J. (1968).
[2] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, Springer, New York, 1981. https://link.springer.com/book/9781461381327.
[3] M.H. Stone, The Theory of Representation for Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111. https://doi.org/10.2307/1989664.
[4] U.M. Swamy, Representation of Universal Algebras by Sheaves, Proc. Amer. Math. Soc. 45 (1974), 55–58.
[5] U.M. Swamy, V.V.R. Rao, Triple and Sheaf Representations of Stone Lattices, Algebra Univ. 5 (1975), 104–113. https://doi.org/10.1007/bf02485239.
[6] U.M. Swamy, G.C. Rao, Almost Distributive Lattices, J. Aust. Math. Soc. A. 31 (1981), 77–91. https://doi.org/10.1017/s1446788700018498.
[7] U.M. Swamy, C.S.S. Raj, R. Chudamani, On Almost Boolean Algebras and Rings, Int. J. Math. Arch. 7 (2016), 23–29.
[8] U.M. Swamy, C.S.S. Raj, R. Chudamani, Annihilators and maximisors in ADL’S, Int. J. Math. Comput. Res. 5 (2017), 1770–1782.