A note on orthodox additive inverse semirings

M. K. Sen; S. K. Maity

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2004)

  • Volume: 43, Issue: 1, page 149-154
  • ISSN: 0231-9721

Abstract

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We show in an additive inverse regular semiring ( S , + , · ) with E • ( S ) as the set of all multiplicative idempotents and E + ( S ) as the set of all additive idempotents, the following conditions are equivalent: (i) For all e , f ∈ E • ( S ) , e f ∈ E + ( S ) implies f e ∈ E + ( S ) . (ii) ( S , · ) is orthodox. (iii) ( S , · ) is a semilattice of groups. This result generalizes the corresponding result of regular ring.

How to cite

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Sen, M. K., and Maity, S. K.. "A note on orthodox additive inverse semirings." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 43.1 (2004): 149-154. <http://eudml.org/doc/32346>.

@article{Sen2004,
abstract = {We show in an additive inverse regular semiring $(S, +, \cdot )$ with $E^\{\bullet \}(S)$ as the set of all multiplicative idempotents and $E^+(S)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e, f \in E^\{\bullet \}(S)$, $ef \in E^+(S)$ implies $fe\in E^+(S)$. (ii) $(S, \cdot )$ is orthodox. (iii) $(S, \cdot )$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.},
author = {Sen, M. K., Maity, S. K.},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {additive inverse semirings; regular semirings; orthodox semirings; additive inverse semirings; regular semirings; orthodox semirings; idempotents},
language = {eng},
number = {1},
pages = {149-154},
publisher = {Palacký University Olomouc},
title = {A note on orthodox additive inverse semirings},
url = {http://eudml.org/doc/32346},
volume = {43},
year = {2004},
}

TY - JOUR
AU - Sen, M. K.
AU - Maity, S. K.
TI - A note on orthodox additive inverse semirings
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2004
PB - Palacký University Olomouc
VL - 43
IS - 1
SP - 149
EP - 154
AB - We show in an additive inverse regular semiring $(S, +, \cdot )$ with $E^{\bullet }(S)$ as the set of all multiplicative idempotents and $E^+(S)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e, f \in E^{\bullet }(S)$, $ef \in E^+(S)$ implies $fe\in E^+(S)$. (ii) $(S, \cdot )$ is orthodox. (iii) $(S, \cdot )$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.
LA - eng
KW - additive inverse semirings; regular semirings; orthodox semirings; additive inverse semirings; regular semirings; orthodox semirings; idempotents
UR - http://eudml.org/doc/32346
ER -

References

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  1. Chaptal N., Anneaux dont le demi groupe multiplicatif est inverse, C. R. Acad. Sci. Paris, Ser. A-B, 262 (1966), 247–277. (1966) Zbl0133.29001MR0190177
  2. Golan J. S.: The Theory of Semirings with Applications in Mathematics, Theoretical Computer Science., Pitman Monographs and Surveys in Pure and Applied Mathematics 54, Longman Scientific, , 1992. (1992) MR1163371
  3. Howie J. M., Introduction to the theory of semigroups., Academic Press, , 1976. (1976) 
  4. Karvellas P. H., Inverse semirings, J. Austral. Math. Soc. 18 (1974), 277–288. (1974) MR0366991
  5. Zeleznekow J., Regular semirings, Semigroup Forum 23 (1981), 119–136. (1981) MR0641993

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