Graded Semigroups

TL;DR

This paper develops a comprehensive theory of graded semigroups, establishing their properties, equivalences, and connections to graded rings and groupoids, with applications to Leavitt path algebras.

Contribution

It introduces the concept of graded semigroups, defines the smash product, and proves key theorems relating graded semigroups to inverse semigroups, Morita equivalence, and graded rings.

Findings

Category isomorphism between S#G-Mod and S-Gr for semigroups with local units

Characterization of strongly graded inverse semigroups via category equivalence

Connection between graded semigroups and graded rings, groupoids, and Leavitt path algebras

Abstract

We systematically develop a theory of graded semigroups, that is semigroups S partitioned by groups G, in a manner compatible with the multiplication on S. We define a smash product S#G, and show that when S has local units, the category S#G-Mod of sets admitting an S#G-action is isomorphic to the category S-Gr of graded sets admitting an appropriate S-action. We also show that when S is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally equivalent to S_1-Mod, where S_1 is the partition of S corresponding to the identity element 1 of G. These results are analogous to well-known theorems of Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded Morita equivalence implies Morita equivalence for semigroups with local units, evincing the wealth of information encoded by the grading of a semigroup. We also give a graded Vagner-Preston theorem, provide numerous examples of naturally-occurring graded semigroups, and explore connections between graded semigroups, graded rings, and graded groupoids. In particular, we introduce graded Rees matrix semigroups, and relate them to smash product semigroups. We pay special attention to graded graph inverse semigroups, and characterise those that produce strongly graded Leavitt path algebras.