On the algebra of semigroup diagrams

Vesna Kilibarda, University of Nebraska - Lincoln

Abstract

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to two natural ways of multiplying semigroup diagrams associated with a given semigroup presentation. In the first part we obtain some new results about the relationship between congruences on free monoids and submonoids of polycyclic monoids. We also investigate the connection between submonoids of the polycyclic monoid and semigroup diagrams, both corresponding to a given monoid presentation. We prove the confluence of the reduction process on semigroup diagrams and introduce a multiplication on semigroup diagrams that "imitates" the multiplication in the polycyclic monoid, leading to an inverse semigroup structure on the set of semigroup diagrams. In the second part, we consider a new type of (partial) multiplication on the set of semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid ${\cal G}\sb{s}$ of semigroup diagrams over the presentation S = $<$ X: R $>$ may be identified with the fundamental groupoid $\gamma(K\sb{s}$) of a certain 2-dimensional complex $K\sb{s}$. Consequently, the vertex groups of the groupoid ${\cal G}\sb{s}$ are isomorphic to the fundamental groups of the complex $K\sb{s}$. Finally, we give some structural information about the fundamental groups of the complex $K\sb{s}$, and calculate them in some interesting cases of semigroup presentations.

Subject Area

Mathematics

Recommended Citation

Kilibarda, Vesna, "On the algebra of semigroup diagrams" (1994). ETD collection for University of Nebraska-Lincoln. AAI9507815.
https://digitalcommons.unl.edu/dissertations/AAI9507815

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