A well-known result in quasigroup theory says that an associative quasigroup is a group, i.e. in quasigroups, associativity forces the existence of an identity element. The converse is, of course, far from true, as there are many, many non-associative loops. However, a remarkable theorem due to David Mumford and C.P. Ramanujam says that any projective variety having a binary morphism admitting a two-sided identity must be a group. Motivated by this result, we define a universal algebra (A; F) to be an MR-algebra if whenever a binary term function m(x, y) in the algebra admits a two-sided identity, then the reduct (A; m(x, y)) must be associative. Here we give some non-trivial varieties of quasigroups, groups, rings, fields and lattices which are MR-algebras. For example, every MR-quasigroup must be isotopic to a group, MR-groups are exactly the nilpotent groups of class 2, while commutative rings and complemented lattices are MR-algebras if and only if they are Boolean.

拟群论中的一个著名结果是结合拟群是一个群，即在拟群中，结合性迫使一个单位元的存在。当然，反之亦然，因为有很多很多非关联循环。然而，David Mumford 和 CP Ramanujam 提出的一个显着定理说，任何具有二元态射的射影簇都必须是一个群。受此结果的启发，我们将全称代数 ( A ;  F )定义为MR 代数，如果当代数中的二元项函数m ( x ,  y ) 承认双边恒等式时，则归约 ( A ;  m (x ,  y )) 必须是关联的。在这里，我们给出了一些非平凡的拟群、群、环、域和格，它们是 MR 代数。例如，每个 MR 准群必须是一个群的同位素，MR 群正是第 2 类的幂零群，而交换环和补格是 MR 代数当且仅当它们是布尔的。