Congruences and Boolean filters of quasi-modular p-algebras

Abd El-Mohsen Badawy; K.P. Shum

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 1, page 109-123
  • ISSN: 1509-9415

Abstract

top
The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set F B ( L ) = [ F ₐ ) : a ∈ B ( L ) is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.

How to cite

top

Abd El-Mohsen Badawy, and K.P. Shum. "Congruences and Boolean filters of quasi-modular p-algebras." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 109-123. <http://eudml.org/doc/270468>.

@article{AbdEl2014,
abstract = {The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set $F_\{B\}(L) = \{[Fₐ): a ∈ B(L)\}$ is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.},
author = {Abd El-Mohsen Badawy, K.P. Shum},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {p-algebras; quasi-modular p-algebras; Boolean filters; direct products; congruences},
language = {eng},
number = {1},
pages = {109-123},
title = {Congruences and Boolean filters of quasi-modular p-algebras},
url = {http://eudml.org/doc/270468},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Abd El-Mohsen Badawy
AU - K.P. Shum
TI - Congruences and Boolean filters of quasi-modular p-algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 109
EP - 123
AB - The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set $F_{B}(L) = {[Fₐ): a ∈ B(L)}$ is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.
LA - eng
KW - p-algebras; quasi-modular p-algebras; Boolean filters; direct products; congruences
UR - http://eudml.org/doc/270468
ER -

References

top
  1. [1] R. Balbes and A. Horn, Stone lattices, Duke Math. J. 37 (1970) 537-543. doi: 10.1215/S0012-7094-70-03768-3 Zbl0207.02802
  2. [2] R. Balbes and Ph. Dwinger, Distributive Lattices (Univ. Miss. Press, 1975). 
  3. [3] G. Birkhoff, Lattice theory, Amer. Math. Soc., Colloquium Publications, 25, New York, 1967. Zbl0153.02501
  4. [4] G. Grätzer, A generalization on Stone's representations theorem for Boolean algebras, Duke Math. J. 30 (1963) 469-474. doi: 10.1215/S0012-7094-63-03051-5 Zbl0121.26702
  5. [5] G. Grätzer, Lattice Theory, First Concepts and Distributive Lattice (W.H. Freeman and Co., San-Francisco, 1971). Zbl0232.06001
  6. [6] G. Grätzer, General Lattice Theory (Birkhäuser Verlag, Basel and Stuttgart, 1978). Zbl0436.06001
  7. [7] O. Frink, Pseudo-complments in semi-lattices, Duke Math. J. 29 (1962) 505-514. doi: 10.1215/S0012-7094-62-02951-4 Zbl0114.01602
  8. [8] T. Katriŭák and P. Mederly, Construction of p-algebras, Algebra Universalis 4 (1983) 288-316. Zbl0536.06004
  9. [9] M. Sambasiva Rao and K.P. Shum, Boolean filters of distributive lattices, Int. J. Math. and Soft Comp. 3 (2013) 41-48. 
  10. [10] P.V. Venkatanarasimhan, Ideals in semi-lattices, J. Indian. Soc. (N.S.) 30 (1966) 47-53. Zbl0158.01604

NotesEmbed ?

top

You must be logged in to post comments.