Trace- and pseudo-products: restriction-like semigroups with a band of projections (Q2665920)

The classic Ehresmann-Schein-Nambooripad (ESN) theorem identifies the category of inverse semigroups with the category of inductive groupoids and their respective functors. In this article, the version of this theorem for the larger class of restriction semigroups is further extended to \textit{localisable semigroups}, semigroups \(S\) with two additional unary operations, \(^+\) and \(^*\) (where we have used the latter notation instead of that of the authors since it is the one readers will be familiar with). They satisfy \(x^+ x = x\), \((xy)^+ = (xy^+)^+\) and \(x^+y^+ = (x^+ y)^+\), along with the ``dual'' identities for \(^*\) and the usual identities that guarantee that the ``left'' projections coincide with the ``right'' projections. Now the projections form a band, rather than a semilattice as in restriction semigroups. In respect of the ESN theorem, the key definition is that of \textit{trace products}, which are defined precisely when \(x^* = y^+\). (For inverse semigroups, this reduces to the usual \(x^{-1}x = y y^{-1}\).) So in some sense these semigroups are optimal for when this definition applies. Then it is shown that the category of localisable semigroups is equivalent to the category of ``transcription categories'', small categories equipped with pairs of ``transcription maps'' satisfying certain axioms. The authors complete the paper with various special cases, including that where the categories are groupoids and the corresponding localisable semigroups are equipped with a compatible regular unary operation.