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By a classical result of Oyestein Ore [5], a domain satisfying a polynomial identity is right Ore, hence has a unique skew field of fractions. Subsequently, it was observed [4] that this "quotient construction" applies to semigroups as well: a cancellation semigroup satisfying a non-trivial semigroup law can be embedded in a group. Thus the following question is natural: if a semigroup S satisfies some non-trivial semigroup laws, must the group of quotients (obtained through Ore's construction) also satisfy these laws? This was raised independently by several authors including G. M. Bergman [1] (and by Padmanabhan in the mid 1980's). In [3] and [4], Mal'cev and B.H. Neumann independently gave a class of examples of semigroup laws having this property. In 2005, this question was answered in the negative by Ivanov and Storozev. [2]. In Part I of this presentation, we explore Ore's quotient condition in the context of semigroups. In Part II, we produce several semigroup laws (and also some equationally defined properties like semigroups admitting conjugates) which are preserved under the quotient construction.

1. Bergman, George M., Hyperidentities of groups and semigroups, Aequationes Math., 23 (no. 1), 50–65, (1981).
2. Ivanov, S. V. and Storozev, A.M. On identities in groups of fractions of cancellative semigroups, Proc. Amer. Math. Soc., 133, 1873–1979 (2005). MR 2137850 (2006c:20056)
3. A.I. Mal'tsev, On the immersion of an algebraic ring into a field Math. Ann., 113, 686–691 (1937). MR1513116A.I.
4. B.H. Neumann and T. Taylor, Semigroups of nilpotent groups, Proc. Royal Soc. A , 274, 1–4 (1963). MR0159884 (28 #3100)
5. Oyestein Ore, Linear equations in non-commutative fields. Ann. of Math., (2) 32, 463–477 (1931).