Several varieties of quasigroups obtained from perfect Mendelsohn designs with block size 4 are defined. One of these is obtained from the so‐called directed standard construction and satisfies the law xy ⋅ (y ⋅ xy) = x and another satisfies Stein's third law xy ⋅ yx = y. Such quasigroups which satisfy the flexible law x ⋅ yx = xy ⋅ x are investigated and characterized. Quasigroups which satisfy both of the laws xy ⋅ (y ⋅ xy) = x and xy .yx = y are shown to exist. Enumeration results for perfect Mendelsohn designs PMD(9, 4) and PMD(12, 4) as well as for (nonperfect) Mendelsohn designs MD(8, 4) are given.

中文翻译：

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从块大小为 4 的完美 Mendelsohn 设计构造的拟群

定义了从块大小为 4 的完美 Mendelsohn 设计中获得的几种拟群。其中一个是从所谓的有向标准构造中获得的，并且满足定律 xy ⋅ (y ⋅ xy) = x，另一个满足 Stein 第三定律 xy ⋅ yx = y。对满足弹性律 x ⋅ yx = xy ⋅ x 的此类拟群进行了研究和表征。满足两个定律 xy ⋅ (y ⋅ xy) = x 和 xy .yx = y 的拟群被证明是存在的。给出了完美 Mendelsohn 设计 PMD(9, 4) 和 PMD(12, 4) 以及（非完美） Mendelsohn 设计 MD(8, 4) 的枚举结果。