Semiassociative relation alegbras and inverse property loops

Abstract

Semiassociative relation algebras are among the three varieties of algebras introduced by Maddux (R. Maddux, Some varieties containing relation algebras, Trans. of Amer. Math. Soc., 272, 501-526, 1982.) and are obtained by replacing the associative law (among the conditions of a relation algebra) by a weaker law known as semiassociative law. This thesis mainly investigates the classes of groupoids and multigroupoids whose complex algebras are semiassociative relation algebras and vice versa. We managed to prove that the complex algebra of a groupoid is a semiassociative relation algebra if and only if the groupoid is an inverse property loop (IP loop). We also proved that the complex algebra of a multigroupoid is a semiassociative algebra if and only if the multigroupoid is a polyloop. These results generated enoughinterest in finding the library of small IP loops and hence we obtained the numbers of non-isomorphic IP loops having order up to 13. Since these were obtained by exhaustive enumeration, they are available for inspection. We have also included in this thesis the classification of IP loops into some important subclasses; we established that the smallest non-abelian IP loop with square property is of order 12 and there are 3 of order 12 and only 2 of order 13; the smallest non-associative IP loop that is both flexible and alternative is of order 12 and there are only 2 of order 12 but none of order 13. We also confirmed that the smallest non-associative Steiner loop is of order 10 and that the smallest non-Steiner non-associative C-loop is of order 12. We also listed the non-associative IP loops having Lagrange property and the smallest Hamiltonian non-associative IP loops. It is surprising to note that there are only 25 non-associative abelian IP loops among more than 12,000 small IP loops. It is well known that the IP loops of exponent 2 are exactly the Steiner loops. In this thesis we also managed to count the IP loops of exponent 3 and exponent 5; there are only 66 non-associative IP loops of exponent 3 (64 are of order 13) and only 10 of exponent 5 (all of order 13).