By A. Hartman

Haim Hanani pioneered the thoughts for developing designs and the speculation of pairwise balanced designs, prime on to Wilson's life Theorem. He additionally led the best way within the examine of resolvable designs, overlaying and packing difficulties, latin squares, 3-designs and different combinatorial configurations. The Hanani quantity is a suite of analysis and survey papers on the leading edge of study in combinatorial layout concept, together with Professor Hanani's personal most up-to-date paintings on Balanced Incomplete Block Designs. different components lined comprise Steiner structures, finite geometries, quasigroups, and t-designs.

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**Example text**

X 2 = x 2 , which further implies x 2 = x . y = x . xy. If xy . y = x . xy holds, then ay = a F. E Emtierr 46 implies that a 2 = a . ay = ay . y = ay = a. Finally, we consider the identity y x . y = x . yx. If yx . y = x . yx holds, then ax = a implies that u 2 = ax . a = x * ax = x u which, by cancellation, implies a = x, that is, a* = a. This completes the proof of the lemma. 0 In what follows, we shall make use of a result due to Mullin et al. [MI. 2. A B ( ( 5 , 9, 13, 17, 29, 49}, 1; v ) exists for all positive integers v = 1 (mod 4) with the possible exception of v = 33, 57, 93, 133.

X y = x (4) x *xy=yx (6) (XY * x)Y = x (8) yx . y = x * yx Proof. 2 are actually equivalent. By replacing x by xy in ( y x . y ) y = x , we get ( ( y . x y ) y ) y = xy, and by cancellation, we have ( ( y . x y ) y = x . Conversely, the identity ( y . x y ) y = x implies yx . y = ( y ( ( y * x y ) y ) ) y = y . x y , that is, the identity ( y . xy)y = x implies ( y x . y ) y = x . 2 are conjugate equivalent. For if a quasigroup satisfies the identity ( y . y x ) y = x , then its transpose will satisfy y ( x y .

0 40 F. E Rennet1 3. 3 have been used in the construction and description of orthogonal arrays with interesting conjugacy properties. Indeed, the most conclusive results we have to date regarding the spectra of short conjugate orthogonal identities pertain to those identities associated with certain classes of n2 X 4 orthogonal arrays. 3, and the reader may consult the references for more details. Henceforth, we let J ( u ( x , y ) = v ( x , y ) ) denote the spectrum of the identity u ( x , y ) = v ( x , y ) .