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Multisemigroups with multiplicities and complete ordered semi-rings
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.ORCID iD: 0000-0003-0295-7087
2017 (English)In: Beitraege zur Algebra und Geometrie, ISSN 0138-4821, E-ISSN 2191-0383, Vol. 58, no 2, 405-426 p.Article in journal (Other academic) Published
Abstract [en]

Motivated by the apperance of multisemigroups in the study of additive 2-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be better siutable for applications in higher representation theory.

Place, publisher, year, edition, pages
Berlin: Springer, 2017. Vol. 58, no 2, 405-426 p.
Keyword [en]
Multi-multisemigroup, multisemigroup, 2-category
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-275206DOI: 10.1007/s13366-016-0320-8OAI: oai:DiVA.org:uu-275206DiVA: diva2:899228
Available from: 2016-02-01 Created: 2016-02-01 Last updated: 2017-08-10Bibliographically approved
In thesis
1. On Effective Representations of Path Semigroups and Multi-Multisemigroups
Open this publication in new window or tab >>On Effective Representations of Path Semigroups and Multi-Multisemigroups
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Publisher
31 p.
Series
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2016:2
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-272173 (URN)
Opponent
Supervisors
Available from: 2016-02-01 Created: 2016-01-12 Last updated: 2016-02-01Bibliographically approved
2. Semigroups, multisemigroups and representations
Open this publication in new window or tab >>Semigroups, multisemigroups and representations
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n.

A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative.

A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup.

Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver.

Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero.

Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities).

Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2017. 29 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 102
Keyword
representation theory, semigroups, multisemigroups, category theory, 2-categories
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-327270 (URN)978-91-506-2647-6 (ISBN)
Public defence
2017-09-25, Polhemssalen, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2017-08-31 Created: 2017-08-07 Last updated: 2017-08-31

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Forsberg, Love

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