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 23
meetings per year in different locations around the region, plus occasional
online activities.
The main participant universities are
East Anglia,
HeriotWatt,
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
In early summer 2020 we held a series of weekly eNBSAN seminars
on Zoom. These have now finished for the summer, but we expect to hold
more online activities in September 2020: watch this space or
email Mark Kambites to be added to the
NBSAN mailing list.
 24th June  Igor Dolinka (Novi Sad)  The prefix membership problem for onerelator groups, and its semigrouptheoretical cousins
( abstract
hide abstract  slides )
Abstract.
In early 1930s, Wilhelm Magnus proved his famous and celebrated result that the word problem is decidable for all onerelator
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 onerelator 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 onerelator Lie algebras. Surprisingly, the problem whether
the word problem is decidable for all onerelator 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 onerelator inverse monoids by showing that the (conjectured)
decidability of the word problem for onerelator 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 onerelator 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 onerelator groups  and thus having implications for the word problem for special
onerelator inverse monoids. These results will be formulated in terms of the classical constructions in combinatorial group theory:
amalgamated free products and HNNextensions.
 1st July  Stuart Margolis (BarIlan) 
Degree 2 Transformation Semigroups as Continuous Maps on
Graphs: Foundations and Structure
( abstract
hide abstract  slides )
Abstract.
We develop the theory of transformation semigroups that have degree 2, that
is,
act by partial functions on a finite set such that the inverse image of
each point has
at most two elements. We show that the graph of fibers of such an action
gives
a deep connection between semigroup theory and graph theory. It is known
that
the KrohnRhodes complexity of a degree 2 action is at most 2. We show that
the monoid of continuous maps on a graph is the translational hull of an
appropriate 0simple semigroup. We show how group mapping semigroups can be
considered as regular covers of their right letter mapping image and
relate this to their graph of fibers.
This is joint work with John Rhodes.
 8th July  Yingying Feng (Foshan)  Min network of congruences on an inverse semigroup
( abstract
hide abstract  slides )
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 LiMin
Wang, Lu Zhang, HaiYuan Huang and ZhiYong Zhou.
 15th July  Robert Gray (East Anglia)
 Solving equations in onerelator monoids
( abstract
hide abstract  slides )
Abstract.
By the Diophantine problem we mean the algorithmic problem of
determining if any given system of equations has a solution. Two classical
results due to Makanin (1977, 1983) show that the Diophantine problem is
decidable in any free monoid, and in any free group. It is natural to ask
to what extent these results can be extended to classes of monoids and
groups which are "close to being free", in some sense. In this talk I will
present some results from joint work with Albert Garreta (Bilbao) in which
we investigated this question for onerelator monoids.
 22nd July  Benjamin Steinberg (City College of New York)  Homological finiteness of onerelator monoids and related monoids
( abstract
hide abstract  slides )
Abstract.
The word problem for onerelator monoids is a longstanding open question.
Kobayashi asked in 2000 whether every onerelator monoid admits a finite
complete rewriting system. A necessary condition to have a finite complete
rewriting system is to satisfy the homological finiteness condition
\( FP_{\infty} \). Kobayashi also asked in 2000 whether every onerelator
monoid is of type \( FP_{\infty} \) and proved that each onerelator monoid
is of type \( FP_3 \).
In this talk, we discuss our proof that every onerelator monoid is
indeed of type \( FP_{\infty} \). Our techniques combine methods from
algebraic topology (monoids acting on CW complexes), homological
algebra and the theory of monoid van Kampen diagrams. This is joint
work with Bob Gray.
 29th July  Marianne Johnson (Manchester)  Tropical representations and identities of plactic monoids
( abstract
hide abstract  slides )
Abstract.
I will report on some joint work with Mark Kambites
(https://arxiv.org/abs/1906.03991). We show that the plactic monoid of
every finite rank has a faithful representation by upper triangular
matrices over the tropical semiring. This answers a question first posed
by Izhakian and subsequently studied by several authors. A consequence is
a proof of a conjecture of Kubat and
Okniński that every plactic monoid of
finite rank satisfies a nontrivial semigroup identity. In the converse
direction, we show that every identity satisfied by the plactic monoid of
rank \( n \) is satisfied by the monoid of \( n \times n \) upper triangular tropical
matrices. In particular this implies that the variety generated by the
\( 3 \times 3 \)
upper triangular tropical matrices coincides with that generated by the
plactic monoid of rank \( 3 \).
Future Meetings
There are currently no physical meetings scheduled, due to the uncertainty
surrounding the COVID19 situation. Meetings will resume as soon as possible.
We expect that there will be further eNBSAN activities (see above) in September 2020; these will be announced here and to the mailing list nearer
the time.
Past Meetings
For the archive of information about past meetings look here.
Other (nonNBSAN) Events and Semigroup News

A fully funded PhD position, which could be in semigroup
theory supervised by
Catarina Carvalho or Yann Peresse, is available at the University of
Hertfordshire. Look
here for more details about the position
or here for information about the algebra
group.
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