On the lattice of congruences on inverse semirings

Anwesha Bhuniya; Anjan Kumar Bhuniya

Discussiones Mathematicae - General Algebra and Applications (2008)

  • Volume: 28, Issue: 2, page 193-208
  • ISSN: 1509-9415

Abstract

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Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences ρ m i n , ρ m a x , ρ m i n and ρ m a x on S and showed that ρ θ = [ ρ m i n , ρ m a x ] and ρ κ = [ ρ m i n , ρ m a x ] . Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ρ m a x is a distributive lattice congruence and ρ m a x is a skew-ring congruence on S. If η (σ) is the least distributive lattice (resp. skew-ring) congruence on S then η ∩ σ is the least Clifford congruence on S.

How to cite

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Anwesha Bhuniya, and Anjan Kumar Bhuniya. "On the lattice of congruences on inverse semirings." Discussiones Mathematicae - General Algebra and Applications 28.2 (2008): 193-208. <http://eudml.org/doc/276932>.

@article{AnweshaBhuniya2008,
abstract = {Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences $ρ_\{min\}, ρ_\{max\}, ρ^\{min\}$ and $ρ^\{max\}$ on S and showed that $ρθ = [ρ_\{min\},ρ_\{max\}]$ and $ρκ = [ρ^\{min\},ρ^\{max\}]$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_\{max\}$ is a distributive lattice congruence and $ρ^\{max\}$ is a skew-ring congruence on S. If η (σ) is the least distributive lattice (resp. skew-ring) congruence on S then η ∩ σ is the least Clifford congruence on S.},
author = {Anwesha Bhuniya, Anjan Kumar Bhuniya},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {inverse semirings; trace; kernel; Clifford congruence; least Clifford congruence; kernel trace approach; Clifford congruences; lattices of congruences},
language = {eng},
number = {2},
pages = {193-208},
title = {On the lattice of congruences on inverse semirings},
url = {http://eudml.org/doc/276932},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Anwesha Bhuniya
AU - Anjan Kumar Bhuniya
TI - On the lattice of congruences on inverse semirings
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2008
VL - 28
IS - 2
SP - 193
EP - 208
AB - Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences $ρ_{min}, ρ_{max}, ρ^{min}$ and $ρ^{max}$ on S and showed that $ρθ = [ρ_{min},ρ_{max}]$ and $ρκ = [ρ^{min},ρ^{max}]$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_{max}$ is a distributive lattice congruence and $ρ^{max}$ is a skew-ring congruence on S. If η (σ) is the least distributive lattice (resp. skew-ring) congruence on S then η ∩ σ is the least Clifford congruence on S.
LA - eng
KW - inverse semirings; trace; kernel; Clifford congruence; least Clifford congruence; kernel trace approach; Clifford congruences; lattices of congruences
UR - http://eudml.org/doc/276932
ER -

References

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