Title:

Constructing monogenic quasigroups with specified properties

A monogenic quasigroup is one generated by a single element, and as such is not just nonassociative but in general non power associative. We show that monogenic quasigroups or loops with various specified characteristics can exist, by demonstrating constructions and by giving examples. This often involves completing an associated partial latin square, or demonstrating that a completion is possible. It is shown that for every order n ≥ 4 there are monogenic quasigroups generated by each of any m ≤ n of their elements, and similarly for monogenic loops (n ≥ 6 , 2 ≤ m ≤ n −1). Any element in a quasigroup must have its powers unambiguous and distinct (called good) up to some degree j ≥ 2, and unambiguous but not necessarily distinct (called clear) up to some degree k ≥ j. The conditions for the existence of a quasigroup of order n having a generator with a good j th and clear k th power are determined. A monogenic quasigroup may be said to be ggood if every element has a good g th power. An algorithm for finding examples based on diagonally cyclic latin squares is developed, and a computer program used to find comprehensive solutions for g ≤ 16 and odd orders n ≤ 95 (and patchily to g = 17, n = 111), with particular reference to the lowest n affording a solution for any g. A maximally non power associative quasigroup has every element with all its bracketings up to some length distinct. A diagonally cyclic quasigroup of order 23 with all 23 products of length ≤ 5 distinct for every element is displayed,as is one of order 63 with 63 of the 65 bracketings up to length 6 distinct for each element. Properties of direct products of monogenic quasigroups, and the significance of parastrophy and isotopy, are considered. The existence or not of monogenic versions of particular types of quasigroups and loops (for example, totally symmetric, inverse property, entropic, Bol and Moufang, among others) is also explored.
