Memoirs of the American Mathematical Society 2000; 89 pp; softcover Volume: 144 ISBN10: 0821820214 ISBN13: 9780821820216 List Price: US$51 Individual Members: US$30.60 Institutional Members: US$40.80 Order Code: MEMO/144/684
 Homogeneous integral table algebras of degree three with a faithful real element. The algebras of the title are classified to exact isomorphism; that is, the sets of structure constants which arise from the given basis are completely determined. Other results describe all possible extensions (preimages), with a faithful element which is not necessarily real, of certain simple homogeneous integral table algebras of degree three. On antisymmetric homogeneous integral table algebras of degree three. This paper determines the homogeneous integral table algebras of degree three in which the given basis has a faithful element and has no nontrivial elements that are either real (symmetric) or linear, and where an additional hypothesis is satisfied. It is shown that all such bases must occur as the set of orbit sums in the complex group algebra of a finite abelian group under the action of a fixedpointfree automorphism of order three. Homogeneous integral table algebras of degree three with no nontrivial linear elements. The algebras of the title which also have a faithful element are determined to exact isomorphism. All of the simple homogeneous integral table algebras of degree three are displayed, and the commutative association schemes in which all the nondiagonal relations have valency three and where some relation defines a connected graph on the underlying set are classified up to algebraic isomorphism. Readership Graduate students and research mathematicians interested in commutative rings and algebras. Table of Contents Part I. Homogeneous Integral Table Algebras of Degree Three with a Faithful Real Element, H. I. Blau and B. Xu  Introduction
 Known facts and some consequences
 Homogeneous ITA's of arbitrary degree
 Some results on bases with a standard quotient
 Extensions of \(\mathbf T_n(3), n>1\)
 Extensions of \(\mathbf V_3\)
 Extensions of \(\mathbf V_2\) and \(\mathbf V_4\)
 Extensions of \(\mathbf T_0(3)\)
 Proof of the main theorem
 References
Part II. On Antisymmetric Homogeneous Integral Table Algebras of Degree Three, Z. Zrad, E. Fisman, V. Miloslavsky, and M. Muzychuk  Introduction
 General facts
 The universal covering
 Perfect triples
 References
Part III. Homogeneous Integral Table Algebras of Degree Three With No Nontrivial Linear Elements, H. I. Blau  Introduction
 Tame and wild elements
 The array
 The cover
 Proofs of Theorem A and Corollary C
 Association schemes
 References
