Department of Mathematics

A semigroup law \( f=g \) is said to be transferable if whenever a cancellation semigroup \( S \) satisfies the identity then \( S\,S^{-1} \) its group of right quotients, also satisfies \( f=g \). For example, the commutative law \( xy = yx \) is well-known to be transferable.

In 1986, G. Bergman in his Berkeley Algebra seminar asked whether every semigroup law is transferable. A similar question was raised independently in the Manitoba Universal Algebra seminar by Padmanabhan as the so-called CS-conjecture. The answer turns out to be negative as proved by Ivanov and Storozhev in 2005. In other words, not every semigroup law is transferable. This raises the important question of finding all transferable semigroup laws. In [6] and [7], Neumann and Mal'cev proved independently that nilpotent laws like \( xyzyx = yxzxy \) are transferable. More transferable laws are given in [4] and [5].

Here we describe an actual quotient construction procedure to demonstrate the transferability of several two-variable laws involving power-like maps.

This research was done in collaboration with G.I. Moghaddam, Yang Zhang and Yi Lin, all of the University of Manitoba.

References:

1. Bergman G.; Hyperidentities of groups and semigroups, Aequat. Math. 23 (1981), 55-65.
2. Bergman G.; Questions in Algebra, Preprint, Berkeley, U.D. 1986.
3. Ivanov S.V., Storozhev A.M.; On identities in groups of fractions of cancellative semigroups, Proc. Amer. Math. Soc. 133 (2005), 1873-1879.
4. Jan Krempa and Olga Macedonska; On identities of cancellation semigroups, Contemporary Mathematics, Vol 131, 1992.
5. Olga Macedonska and Piotr Slanina; On identities satisfied by cancellative semigroups and their groups of fractions, (Preprint)
6. A.I. Mal’cev; Nilpotent Groups, Ivanov Gos.Ped.Ins.Uc.zap (1953).
7. B.H. Neumann and T. Taylor; Subsemigroups of Nilpotent Groups, Proc.Roy.Soc.Ser.A, 274(1963), pp 1-4.
8. G. I. Moghaddam and R. Padmanabhan; Cancellative Semigroups Admitting Conjugates, J. Semigroup Theory 9 (2015)
9. R. Padmanabhan and Yang Zhang; Commutativity Theorems in Groups with Power-like Maps, J. Formalized Reasoning 12 (2019) No, 1, 1-10.