Projection algebras and free projection- and idempotent-generated regular $*$-semigroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908310506045440 |
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| author | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik |
| author_facet | East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik |
| contents | 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 $P\mapsto PG(P)$ 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 theory is then 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 $P(TL_n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09109 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection algebras and free projection- and idempotent-generated regular $*$-semigroups East, James Gray, Robert D. Muhammed, P. A. Azeef Ruškuc, Nik Rings and Algebras Category Theory Group Theory 20M50 (primary), 18B40, 20M05, 20M10, 20M17, 20M20 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 $P\mapsto PG(P)$ 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 theory is then 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 $P(TL_n)$. |
| title | Projection algebras and free projection- and idempotent-generated regular $*$-semigroups |
| topic | Rings and Algebras Category Theory Group Theory 20M50 (primary), 18B40, 20M05, 20M10, 20M17, 20M20 |
| url | https://arxiv.org/abs/2406.09109 |