considered an involuted semigroup with operation given by ab = [a, e, b] and involution by a–1 = [e, a, e]. Theorem: Every semiheap may be embedded in an involuted semigroup. As in the study of semigroups, the structure of semiheaps is described in terms of ideals with an "i-simple semiheap" being one with no proper ideals. Mustafaeva translated the Green's relations of semigroup theory to semiheaps and defined a ρ class to be those elements generating the same principle two-sided ideal. He then proved that no i-simple semiheap can have more than two ρ classes. He also described regularity classes of a semiheap S: where n and m have the same parity and the ternary operation of the semiheap applies at the left of a string from S. He proves that S can have at most 5 regularity classes. Mustafaev calls an ideal B "isolated" when He then proves that when S = D(2,2), then every ideal is isolated and conversely. Studying the semiheap Z(A, B) of heterogeneous relations between sets A and B, in 1974 K. A. Zareckii followed Mustafaev's lead to describe ideal equivalence, regularity classes, and ideal factors of a semiheap. Generalizations and related concepts A pseudoheap or pseudogroud satisfies the partial para-associative condition A Malcev operation satisfies the identity law but not necessarily the para-associative law, that is, a ternary operation on a set satisfying the identity . A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law. An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with where • denotes matrix multiplication and T denotes matrix transpose. An idempotent semiheap is a semiheap where for all a. A generalised heap or generalised groud is an idempotent semiheap where and for all a and b. A semigroud is a generalised groud if the relation → defined by is reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation. See also n-ary associativity Notes References Anton Sushkevich (1929) "On a generalization of the associative law", Transactions of the American Mathematical Society 31(1): 204–14 External links Non-associative algebra Ternary operations