On Union of Regular Near-rings

TL;DR

This paper extends the structure theorem for seminearrings by characterizing those that are unions of various types of regular near-rings, generalizing previous results on completely regular seminearrings.

Contribution

It introduces a characterization of seminearrings as unions of different classes of regular near-rings, broadening the understanding of their algebraic structure.

Findings

Seminearrings can be expressed as unions of regular near-rings.

Characterization of seminearrings as unions of inverse and Clifford near-rings.

Extension of structure theorems from completely regular to various regular near-rings.

Abstract

'A semigroup is completely regular if and only if it is a union of groups'- an analogue of this structure theorem of completely regular semigroup has been obtained in the setting of seminearrings in [[16], Mukherjee (Pal) et al., Semigroup Forum (2018)]. In it, a class of seminearrings (called generalized left completely regular seminearrings, abbreviated as GLCR) has been characterized as a union of near-rings. This work has been extended in the present article to characterize the seminearrings which are union of various types (regular, completely regular, inverse, Clifford) of regular near-rings.