Department of Mathematics Server University of Manitoba

Abstract: This conjecture says that for a particular kind of equational statement, if it can be proved within the realms of the equational theory of groups, then it can be proved within the much weaker theory of cancellative semigroups. More precisely,

• Let GT be an equational axiomatization of group theory in terms of a binary product, unary inverse, and a nullary identity element.
• Let CS be the associative law of binary product and the two cancellation laws.
• Let A be a set of equations in the language of one binary operation (product) and variables, and let a=b be an equation of the same kind.
• Then If (A,GT implies "a=b") then (A,CS implies "a=b").

This is really a meta-conjecture and perhaps not true in general. However, there are many CS implications in GT which are provable in "pure" CS. Recently Professor B.H. Neumann has published an article on this topic (to appear in a forthcoming issue of the CMS Bulletin). In this talk, I would like to give an update on this topic and investigate some famous group-theoretical concepts which can be captured within the language of semigroups.