On The Universality Of Central Loops

TL;DR

This paper investigates the properties and conditions under which various types of central loops, such as LC, RC, and C-loops, are universal, G-loops, or Bol loops, revealing their interrelations and specific criteria.

Contribution

It provides new necessary and sufficient conditions for central loops to be universal, G-loops, or Bol loops, expanding understanding of their structural properties.

Findings

LC(RC)-loops are left(right) universal loops.

Universal loops among these are characterized as Bol or Moufang loops.

Conditions for loops and their isotopes to be C-loops or G-loops are established.

Abstract

LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed that not all RC-loops or LC-loops or C-loops are universal loops. But if an RC-loop(LC-loop, C-loop) is universal, then it is a right Bol loop(left Bol loop, Moufang loop) respectively. If a loop and its right or left isotope are commutative then the loop is a C-loop if and only if its right or left isotope is a C-loop. If a C-loop is central square and its right or left isotope is an alternative central square loop, then the latter is a C-loop. Necessary and sufficient condition for an LC-loop(RC-loop) to be a left(right)G-loop is established. Consequently, necessary and sufficient conditions for an LC-loop, and an RC-loop to be a G-loop are established. A necessary and sufficient condition for a C-loop to be a G-loop is established.