k-Congruences on semirings

TL;DR

This paper introduces the concept of k-congruences in semirings, establishes their properties and relationships with k-ideals, and explores conditions for simplicity and zeros, providing new theoretical insights into semiring structure.

Contribution

It defines k-congruences in semirings, proves their correspondence with k-ideals, and clarifies conditions for zero existence and simplicity, correcting a previous lemma.

Findings

K-congruences are in bijection with k-ideals.

A semiring is k-simple iff it is k-congruence-simple.

Inclines are k-simple if they have at most 2 elements.

Abstract

For any semiring, the concept of k-congruences is introduced, criteria for k-congruences are established, it is proved that there is an inclusion-preserving bijection between k-congruences and k-ideals, and an equivalent condition for the existence of a zero is presented with the help of k-congruences. It is shown that a semiring is k-simple iff it is k-congruence-simple, and that inclines are k-simple iff they have at most 2 elements. Lemma 2.12(i) in [Glas. Mat. 42(62) (2007) 301] is pointed out being false.