A note on Birget-Rhodes expansion of groups
Let \(S\) be a semigroup. For \(s_ 1,...,s_ k\in S\) put \(P(s_ 1,...,s_ k)=\{1,s_ 1,s_ 1s_ 2,...,s_ 1s_ 2...s_ k\}\). Define \(\tilde S^ R=\{(P(s_ 1,...,s_ k),s_ 1s_ 2...s_ k)\mid s_ i\in S\), \(k\geq 1\}\). It is known that, with appropriate multiplication, \(\tilde S^ R\) is a semigroup. If \(\phi:S\to T\) is a semigroup homomorphism then defining \(\tilde\phi^ R: \tilde S^ R\to \tilde T^ R\) by \[ (P(s_ 1,...,s_ k),s_ 1...s_ k){\tilde \phi}^ R=(P(s_ 1\phi,...,s_ k\phi),s_ 1\phi...s_ k\phi) \] we get a functor \((\tilde .)R\), called the Birget-Rhodes expansion. Denote by \(\sigma =\sigma_ S\) the least group congruence on \(S\). An inverse semigroup is called F-inverse if every \(\sigma\)-class has a greatest element with respect to the natural partial order. It is proved that \((\tilde .)^ R\) is a functor from the category of groups into the category of F-inverse semigroups and that this functor is a left adjoint of the functor assigning the greatest group homomorphic image to any F-inverse semigroup. An abstract characterization of \(\tilde G^ R\) for any group \(G\) is presented.
scientific article; zbMATH DE number 4107112