University of Birmingham > Talks@bham > Theoretical computer science seminar > Are all substitutions invertible; are all monoids groups?

Are all substitutions invertible; are all monoids groups?

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If you have a question about this talk, please contact Paul Levy.

Clearly, not everything we do in life is reversible. That’s why mathematicians have monoids. However, every monoid can be mapped to a corresponding group in a natural way, so perhaps everything we do in mathematics is reversible after all.

This is not obvious. Consider the two-element monoid with elements {0,1} and 1+1=1. We can naively add an inverse -1 to 1, but then we quickly derive 1=0; the monoid collapses to the trivial group, which is not what we intended. I will show how it is nevertheless possible to get an adjunction between categories of monoids and groups such that, in a suitable sense, every monoid can be injectively mapped into a group, and vice-versa. I will go on to speculate on applications of these constructions to reversible computation, unification, and rewriting.

This is joint work with Peter Kropholler in Glasgow. For more information, see

This talk is part of the Theoretical computer science seminar series.

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