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 x y ⋅ ( y ⋅ x y ) = x and another satisfies Stein's third law x y ⋅ y x = y. Such quasigroups which satisfy the flexible law x ⋅ y x = x y ⋅ x are investigated and characterized. Quasigroups which satisfy both of the laws x y ⋅ ( y ⋅ x y ) = x and x y ⋅ y x = 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.

A directed triple system can be defined as a decomposition of a complete digraph to directed triples x, y, z . By setting xy = z, yz = x, xz = y and uu = u we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.

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We complete the existence spectrum of perfect Mendelsohn designs v PMD(, 5) as v 0, 1 (mod 5), ≡ v 6, 10 ≠ by exhibiting previously unknown designs PMD(15, 5) and PMD(20, 5).

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