By A. Hartman

Haim Hanani pioneered the recommendations for developing designs and the idea of pairwise balanced designs, prime on to Wilson's life Theorem. He additionally led the best way within the learn of resolvable designs, protecting 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 analysis in combinatorial layout conception, together with Professor Hanani's personal most modern paintings on Balanced Incomplete Block Designs. different components coated contain Steiner structures, finite geometries, quasigroups, and t-designs.

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**Example text**

Obviously, J ( ( x . y x ) y = x ) = J ( y x . x y = x ) . 6. There exists an idempotent quasigroup of order n satisfying Stein's third law for all positive integers n = 0 or 1 (mod 4) except n = 4, and possibly excepting n = 8, 12, 33. Remark. K. Heinrich [private communication] has informed the author that an exhaustive computer search established the non-existence of a (8,4, 1)-perfect Mendelsohn design. 6. :E Bennett 42 the identity ( x y y ) y = x and n 2 x 4 orthogonal arrays having C 3 (the cyclic group of order 3) as conjugate invariant subgroup.

110, 114, 126, 130, 142, 158, 162, 174, 178, 190). 4. For all integers u 3 4 , u E B(4, 5, 9, 11, 19, 31) holds with the exception of u E (6, 7, 8, 10, 12, 14, 15, 18, 22, 23, 26, 27, 30, 34) and with the possible exception of u E (38, 42, 43, 46, 50, 54, 62, 66, 70, 74, 78, 82, 90, 98, 102, 114, 126). 3. 5. u E J ( x . xy = y x ) holds for all positive integers u except u E (2, 3, 6, 7, 8, 10, 12, 14) and possibly excepting u E (15, 18, 22, 23, 26, 27, 30, 34, 38, 42, 43, 46, 50, 54, 62, 66, 70, 74, 78, 82, 90, 98, 102, 114, 126).

W), if u f w . The quasigroup ( S , @) SO constructed is called the singular direct product of V and Q. Unlike the direct product construction, two-variable quasigroup identities are not necessarily preserved by the singular direct product construction. C. Lindner [41] has obtained some fairly general results on identities which are preserved by the singular direct product for quasigroups. Before stating the result, we need to adapt some of the terminology used in [41]. Let F ( x , y) be the free groupoid on two generators x and y.