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ToolsEast, James, Gray, Robert D., Muhammed, P. A. Azeef and Ruškuc, Nik (2025) Projection algebras and free projection- and idempotent-generated regular ⁎-semigroups. Advances in Mathematics, 473. ISSN 0001-8708
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Abstract
The purpose of this paper is to introduce a new family of semigroups—the free projection-generated regular ⁎-semigroups—and initiate their systematic study. Such a semigroup PG(P) is constructed from a projection algebra P, using the recent groupoid approach to regular ⁎-semigroups. The assignment P↦PG(P) is a left adjoint to the forgetful functor that maps a regular ⁎-semigroup S to its projection algebra P(S). In fact, the category of projection algebras is coreflective in the category of regular ⁎-semigroups. The algebra P(S) uniquely determines the biordered structure of the idempotents E(S), up to isomorphism, and this leads to a category equivalence between projection algebras and regular ⁎-biordered sets. As a consequence, PG(P) can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups IG(E) and RIG(E), where E=E(PG(P)); this is witnessed by a number of presentations in terms of generators and defining relations. The semigroup PG(P) can also be interpreted topologically, through a natural link to the fundamental groupoid of a simplicial complex explicitly constructed from P. The above theory is illustrated on a number of examples. In one direction, the free construction applied to the projection algebras of adjacency semigroups yields a new family of graph-based path semigroups. In another, it turns out that, remarkably, the Temperley–Lieb monoid TL n is the free regular ⁎-semigroup over its own projection algebra P(TL n).
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | chained projection groupoid,free idempotent-generated semigroup,free projection-generated regular ⁎-semigroup,fundamental groupoid,projection algebra,regular ⁎-semigroup,mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600 |
| Faculty \ School: | Faculty of Science > School of Engineering, Mathematics and Physics |
| UEA Research Groups: | Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR) |
| Related URLs: | |
| Depositing User: | LivePure Connector |
| Date Deposited: | 10 Apr 2025 10:30 |
| Last Modified: | 15 May 2025 09:30 |
| URI: | https://ueaeprints.uea.ac.uk/id/eprint/99014 |
| DOI: | 10.1016/j.aim.2025.110288 |
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