# North British Semigroups and Applications Network

NBSAN is a network of researchers in Scotland and Northern England with
interests in semigroup theory and its applications. It is funded chiefly by a grant
from the London
Mathematical Society (with additional contributions from host departments
and occasionally from other sources)
and holds 2-3
meetings per year in different locations around the region.
The main participant universities are
East Anglia,
Heriot-Watt,
Manchester,
St Andrews
and York, but our
activities are open to all interested researchers. Attendance at all meetings
is free. We especially welcome
graduate students.

To enquire about the network, or to join the mailing list, please contact
the central coordinator Mark Kambites. For enquiries about individual meetings,
please contact the relevant local organiser as described below.

## eNBSAN

We will be holding a series of weekly **eNBSAN seminars on Zoom** during the summer,
hosted by the various NBSAN participant universities.
Seminars will take place on **Wednesdays at 1pm British Summer Time**
(**12 noon GMT/UTC**), starting on 24th June. The schedule so
far:

- 24th June -
**Igor Dolinka** (Novi Sad) - *The prefix membership problem for one-relator groups, and its semigroup-theoretical cousins*
(abstract)
(hide abstract)
**Abstract.**
In early 1930s, Wilhelm Magnus proved his famous and celebrated result that the word problem is decidable for all one-relator
groups: given two group words \( u,v \) over the alphabet \( X \), there is an algorithm deciding whether \( u \) and \( v \) represent the
same element of the group given by a presentation of the form \( \langle X \mid w=1 \rangle \). This result is based on another important theorem
proved earlier by Magnus, the *Freiheitssatz*, which, roughly speaking, locates many free subgroups in one-relator groups.
Later on, this inspired investigations of the word problem for other algebraic structures defined by a single relation. For example,
in the 1960s Shirshov proved that the word problem is decidable for all one-relator Lie algebras. Surprisingly, the problem whether
the word problem is decidable for all one-relator monoids is still open (although several important cases have been resolved by
Adjan in 1966, and Adyan and Oganessyan in 1987).

An important intermediate class of algebraic structures lying between groups and monoids are that of inverse monoids. In 2001
Ivanov, Margolis and Meakin highlighted the importance of investigating one-relator inverse monoids by showing that the (conjectured)
decidability of the word problem for one-relator *special* inverse monoids (the ones defined by a relation of the form \( w=1 \))
would imply a positive solution of the word problem for all one-relator monoids. They also showed that, under a condition that is
very familiar within the inverse semigroup theory realm, in the *\( E \)-unitary case*, solving the word problem for the inverse
monoid \( M \) given by the presentation \( \langle X \mid w=1 \rangle \) is equivalent to solving the *prefix membership problem* for the group \( G \)
given by the same presentation. This problem asks whether there exists an algorithm deciding whether a given word \( u \) represents
an element of the submonoid \( P_w \) of \( G \) generated by all prefixes of the relator word \( w \).

In this talk I will present several recent results, obtained in collaboration with Robert D. Gray (UEA Norwich), pertaining to the
prefix membership problem for a class of one-relator groups -- and thus having implications for the word problem for special
one-relator inverse monoids. These results will be formulated in terms of the classical constructions in combinatorial group theory:
amalgamated free products and HNN-extensions.

- 1st July -
**Stuart Margolis** (Bar Ilan)
- 8th July -
**Yingying Feng** (Foshan) - *Min network of congruences on an inverse semigroup*
(abstract)
(hide abstract)
**Abstract.** A congruence on an inverse semigroup S is determined uniquely by its kernel
and trace. Denoting by \( \rho_k \) and \( \rho_t \) the least congruence on S
having the same kernel and the same trace as \( \rho \), respectively, and
denoting by \( \omega \) the universal congruence on S, we consider the sequence
\( \omega, \omega_k, \omega_t, (\omega_k)_t, (\omega_t)_k, \cdots \)

We call these congruences, together with the inclusion relation for
congruences, the min network of congruences on S. The quotients
\( \{S/\omega_k\}, \{S/\omega_t\}, \{S/(\omega_k)_t\}, \{S/(\omega_t)_k\},
\cdots \), as S runs over all inverse semigroups, form quasivarieties.

In this talk, I will talk about the repeated patterns in the resulting
quotient semigroups. These patterns help us not only determine the
quasivarieties to which the quotient semigroups belong, but also obtain
relationships among these quasivarieties. This is joint work with Li-Min
Wang, Lu Zhang, Hai-Yuan Huang and Zhi-Yong Zhou.

- 15th July -
**Robert Gray** (East Anglia)
- 22nd July -
**Benjamin Steinberg** (City College of New York)
- 29th July -
**Marianne Johnson** (Manchester)

More details will appear here, and be announced to the NBSAN mailing
list, nearer the time. For security, the meeting link and password
will be circulated only to the mailing list. In the meantime please
contact Mark Kambites with any questions or
to be added to the mailing list.

## Future Meetings

There are currently no physical meetings scheduled, due to the uncertainty
surrounding the COVID-19 situation. Meetings will resume as soon as possible.

## Past Meetings

For the archive of information about past meetings look here.

This page is maintained by

Mark Kambites.
It was retrieved on 3rd June 2020.
The text was last manually edited on 2nd June 2020 but dynamically
generated content may have changed more recently.

Opinions expressed are those of the author and do not necessarily reflect policy of the University of Manchester or any other organisation.

Information is correct to the best of the author's knowledge but is provided without warranty.

All content is protected by copyright, and may not be reproduced or further distributed without permission.