Distributive lattices of t-k-Archimedean semirings

Tapas Kumar Mondal

Discussiones Mathematicae - General Algebra and Applications (2011)

  • Volume: 31, Issue: 2, page 147-158
  • ISSN: 1509-9415

Abstract

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A semiring S in 𝕊𝕃⁺ is a t-k-Archimedean semiring if for all a,b ∈ S, b ∈ √(Sa) ∩ √(aS). Here we introduce the t-k-Archimedean semirings and characterize the semirings which are distributive lattice (chain) of t-k-Archimedean semirings. A semiring S is a distributive lattice of t-k-Archimedean semirings if and only if √B is a k-ideal, and S is a chain of t-k-Archimedean semirings if and only if √B is a completely prime k-ideal, for every k-bi-ideal B of S.

How to cite

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Tapas Kumar Mondal. "Distributive lattices of t-k-Archimedean semirings." Discussiones Mathematicae - General Algebra and Applications 31.2 (2011): 147-158. <http://eudml.org/doc/276484>.

@article{TapasKumarMondal2011,
abstract = {A semiring S in 𝕊𝕃⁺ is a t-k-Archimedean semiring if for all a,b ∈ S, b ∈ √(Sa) ∩ √(aS). Here we introduce the t-k-Archimedean semirings and characterize the semirings which are distributive lattice (chain) of t-k-Archimedean semirings. A semiring S is a distributive lattice of t-k-Archimedean semirings if and only if √B is a k-ideal, and S is a chain of t-k-Archimedean semirings if and only if √B is a completely prime k-ideal, for every k-bi-ideal B of S.},
author = {Tapas Kumar Mondal},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {k-radical; t-k-Archimedean semiring; completely prime k-ideal; semiprimary k-ideal; -radical, --Archimedean semirings; completely prime -ideals; semiprimary -ideals},
language = {eng},
number = {2},
pages = {147-158},
title = {Distributive lattices of t-k-Archimedean semirings},
url = {http://eudml.org/doc/276484},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Tapas Kumar Mondal
TI - Distributive lattices of t-k-Archimedean semirings
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2011
VL - 31
IS - 2
SP - 147
EP - 158
AB - A semiring S in 𝕊𝕃⁺ is a t-k-Archimedean semiring if for all a,b ∈ S, b ∈ √(Sa) ∩ √(aS). Here we introduce the t-k-Archimedean semirings and characterize the semirings which are distributive lattice (chain) of t-k-Archimedean semirings. A semiring S is a distributive lattice of t-k-Archimedean semirings if and only if √B is a k-ideal, and S is a chain of t-k-Archimedean semirings if and only if √B is a completely prime k-ideal, for every k-bi-ideal B of S.
LA - eng
KW - k-radical; t-k-Archimedean semiring; completely prime k-ideal; semiprimary k-ideal; -radical, --Archimedean semirings; completely prime -ideals; semiprimary -ideals
UR - http://eudml.org/doc/276484
ER -

References

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  1. [1] A.K. Bhuniya and K. Jana, Bi-ideals in k-regular and intra k-regular semirings, accepted for publication in Discuss. Math. General Algebra and Applications 31 (2011), 5-25. Zbl1254.16040
  2. [2] A.K. Bhuniya and T.K. Mondal, Distributive lattice decompositions of semirings with a semilattice additive reduct, Semigroup Forum 80 (2010), 293-301. doi: 10.1007/s00233-009-9205-6 Zbl1205.16039
  3. [3] S. Bogdanovic and M. Ciric, Semilattice of Archimedean semigroups and completely π-regular semigroups I (survey article), Filomat(nis) 7 (1993), 1-40. Zbl0848.20052
  4. [4] S. Bogdanovic and M. Ciric, Chains of Archimedean semigroups (Semiprimary semigroups), Indian J. Pure and Appl. Math. 25 (1994), 229-235. Zbl0801.20045
  5. [5] M. Ciric and S. Bogdanovic, Semilattice decompositions of semigroups, Semigroup Forum (1996), 119-132. doi: 10.1007/BF02574089 
  6. [6] A.H. Clifford, Semigroups admitting relative inverses, Annals of Math. 42 (1941), 1037-1049. doi: 10.2307/1968781 Zbl0063.00920
  7. [7] F. Kmet, Radicals and their left ideal analogues in a semigroup, Math. Slovaca 38 (1988), 139-145. Zbl0643.20042
  8. [8] M. Petrich, The maximal semilattice decomposition of a semigroup, Math. Zeitschr. 85 (1964), 68-82. doi: 10.1007/BF01114879 Zbl0124.25801
  9. [9] M.S. Putcha, Semilattice decomposition of semigroups, Semigroup Forum 6 (1973), 12-34. doi: 10.1007/BF02389104 
  10. [10] T. Tamura, Another proof of a theorem concerning the greatest semilattice decomposition of a semigroup, Proc. Japan Acad. 40 (1964), 777-780. doi: 10.3792/pja/1195522562 Zbl0135.04001
  11. [11] T. Tamura, On Putcha's theorem concerning semilattice of archimedean semigroups, Semigroup Forum 4 (1972), 83-86. doi: 10.1007/BF02570773 Zbl0256.20075
  12. [12] T. Tamura, Note on the greatest semilattice decomposition of semigroups, Semigroup Forum 4 (1972), 255-261. doi: 10.1007/BF02570795 Zbl0261.20058
  13. [13] T. Tamura and N. Kimura, On decomposition of a commutative semigroup, Kodai Math. Sem. Rep. 4 (1954), 109-112. doi: 10.2996/kmj/1138843534 Zbl0058.01503

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