On additively regular seminearrings

The most natural seminearrings, i.e., the seminearrings of all self-maps of additive semigroups are necessarily multiplicatively regular but they need not be additively regular. The purpose of this paper is to investigate additively regular seminearrings. We mainly focus on the study of congruences in various types of additively regular seminearrings such as additively inverse, additively Clifford and Bandelt seminearrings. We deduce that for a restricted type of additively inverse seminearrings there exists an inclusion preserving bijective correspondence between the set of all normal congruences and that of all full k-ideals. Finally, we characterize those seminearrings which are the subdirect product of a distributive lattice and a zero symmetric near-ring.