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The University of Southampton
Mathematical Sciences

Research Group: Semigroup Theory

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A semigroup is one of the most fundamental of all abstract algebras having as its basis an associative operation on a non-empty set. We focus on a number of types of semigroups and in particular on actions of semigroups on sets and 'set with structure'.

We are interested in the following specific topics:

  • Actions of inverse semigroups on graphs and trees. The theory of group actions has proved a powerful tool in combinatorial group theory and it is reasonable to expect that useful techniques in semigroup theory may be obtained by trying to `port' the Bass-Serre theory to a semigroup context. Given the importance of transitivity in the group case, we believe it mightreasonably be achieved by restricting our attention to the class of inverse semigroups. However, it very soon becomes apparent that there are some fundamental differences with inverse semigroup actions and even such basic notions such as free actions have to be treated carefully.
  • Actions of E-dense semigroups on sets. Many semigroups originally came to the interest of researchers as natural generalisations of groups. One such type is the class of E-dense semigroups. A semigroup S is said to be E-dense if for every element x in S there exists an element y in S such that xy is idempotent. This project is involved in studying the structure of E-dense acts over E-dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based on considering representations of E-dense semigroups by partial one-to-one maps in a similar way to inverse semigroups. We have discovered applications to the discrete log problem as used in modern cryptography.
  • Adequate transversals of abundant semigroups. The concept of an inverse transversal was introduced in 1978 by Blyth and McAlister and arises in response to natural questions concerning the greatest idempotent in a naturally ordered regular semigroup. At about the same time, Fountain introduced us to abundant and adequate semigroups as very natural generalisations of regular and inverse semigroups. Later El-Qallali, and even later Chen, Luo and others, generalised the notion of transversal to abundant semigroups and it is this generalisation that we consider in this research area. El-Qallali and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. We have been studying a structure theorem for adequate transversals of certain types of quasi-adequate semigroup left ample semigroups and provide a structure for these based on semidirect products of adequate semigroups by certain types of bands.

 The group welcomes applications for postgraduate studies. Please contact group members (email, telephone, or in person) for more information; and when ready, please apply through the Graduate School application page.

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