Near-ring congruences on additively regular seminearrings
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Near‑ring congruences on additively regular seminearrings Kamalika Chakraborty1 · Pavel Pal2 · Sujit Kumar Sardar1 Received: 1 September 2019 / Accepted: 3 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we first obtain analogues of some results of LaTorre (Semigroup Forum 24(1):327–340, 1982) in the setting of additively regular seminearrings which in turn not only give rise to refinements of some important results viz. Propositions 3.16, 3.17, Theorem 3.20 of Sardar and Mukherjee (Semigroup Forum 93(3):629–631, 2016) and Theorem 3.22 of Sardar and Mukherjee (Semigroup Forum 88(3):541– 554, 2014) (involving mainly near-ring congruences i.e., normal congruences and normal full k-ideals of additively inverse seminearrings) but also answer partially a question raised in Sardar and Mukherjee (Semigroup Forum 88(3):541–554, 2014). Finally we study the lattice structures of near-ring congruences and normal full k-ideals in distributively generated additively regular seminearrings. Keywords Additively regular seminearring · Distributively generated seminearring · Near-ring congruence · Normal congruence · Normal full right k-ideal · Normal full ideal
1 Introduction It is well-known that if G is an additive group (not necessarily abelian) then the set M(G) of all mappings from G into G forms a near-ring under point wise addition and composition (cf. [10]). Following [10], we call an algebraic structure (N, +, ⋅) a near-ring if it satisfies the following axioms : (1) (N, +) is a group (not Communicated by Lev Shevrin. * Sujit Kumar Sardar [email protected] Kamalika Chakraborty [email protected] Pavel Pal [email protected] 1
Jadavpur University, Jadavpur, Kolkata 700032, India
2
Bankura University, Bankura 722155, India
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necessarily abelian), (2) (N, ⋅) is a semigroup (not necessarily commutative) and (3) (b + c) ⋅ a = b ⋅ a + c ⋅ a for all a, b, c ∈ N (“right distributive law”). If the group G is replaced by an additive semigroup S (not necessarily commutative) then M(S), with the same operations as that of M(G), no longer forms a near-ring but forms an algebraic structure what is known as seminearring (cf. Definition 1). It is well known that the study of universal algebras in general, and that of semigroups, semirings, seminearrings etc in particular, is heavily dependent on the study of congruences. In this direction group congruences on semigroups, ring congruences on semirings play an important role. Likewise it is natural to study near-ring congruences on seminearrings. In [11], Sardar and Mukherjee studied near-ring congruences on additively inverse seminearrings. Among other results, they obtained for a distributively generated additively inverse seminearring S with the property D (cf. Definition 4) (i) an inclusion preserving bijective correspondence (Theorem 3.20 of [12]) between the set of all near-ring congruences i.e., normal congruences1 on S and the set of all normal full k-i
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