On k-radicals of Green's relations in semirings with a semilattice additive reduct

Mondal, Tapas Kumar; Nath, Bhupendra; Smriti, Dutta; Bhuniya, A. K.

doi:10.7151/dmgaa.1198

Discuss. Math. - General Algebra and Applications

2013

DOI: 10.7151/dmgaa.1198

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On k-radicals of Green's relations in semirings with a semilattice additive reduct

Tapas Kumar Mondal¹,

Bhupendra Nath²,

Dutta Smriti³

et al.

Abstract: We introduce the k-radicals of Green's relations in semirings with a semilattice additive reduct, introduce the notion of left k-regular (right k-regular) semirings and characterize these semirings by k-radicals of Green's relations. We also characterize the semirings which are distributive lattices of left ksimple subsemirings by k-radicals of Green's relations.

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Cited by 3 publications

References 11 publications

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On the least distributive lattice congruence on a semiring with a semilattice additive reduct

Bhuniya

Mondal²

2015

Acta Math. Hungar.

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We characterize the least distributive lattice congruence on the semirings S with semilattice additive reduct in different ways arising from the nonsymmetry and non-transitivity of the binary relation −→ defined by:which are the principal completely semiprime k-ideal and the principal filter of S generated by a respectively. Both the relations M and N induced by M (a) and N (a) respectively are the least distributive lattice congruence on S. Again non-transitivity of −→ yields an expanding family {−→ n } of binary relations which again associates subsets Mn(a) and Nn(a) for all a ∈ S, and induces equivalence relations Mn and Nn. Also the n-th power of the symmetric opening ←→ of −→ gives us Σn(a) = {x ∈ S | a ←→ n x} which induces the equivalence relation σn. We have characterized the semirings which are distributive lattices of Mn(Nn, σn)-simple semirings.

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On the least distributive lattice congruence on a semiring with a semilattice additive reduct

Bhuniya

Mondal²

2015

Acta Math. Hungar.

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On decompositions of semirings via k-radicals of some relations

Bhuniya

Mondal²

2018

Acta Math. Hungar.

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Semirings which are distributive lattices of $t$-$k$-simple semirings

Mondal

Bhuniya

2015

Tbilisi Math. J.

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A semiring S is said to be a t-k-simple semiring if it has no non-trivial proper left k-ideal and no non-trivial proper right k-ideal. We introduce the notion of t-k-simple semirings and characterize the semirings in SL + , the variety of all semirings with a semilattice additive reduct, which are distributive lattices of t-k-simple subsemirings. A semiring S is a distributive lattice of t-k-simple subsemirings if and only if every k-bi-ideal in S is completely semiprime k-ideal. Also the semirings for which every k-bi-ideal is completely prime has been characterized.

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On k-radicals of Green's relations in semirings with a semilattice additive reduct

Cited by 3 publications

References 11 publications

On the least distributive lattice congruence on a semiring with a semilattice additive reduct

On the least distributive lattice congruence on a semiring with a semilattice additive reduct

On decompositions of semirings via k-radicals of some relations

Semirings which are distributive lattices of $t$-$k$-simple semirings

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