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, plus occasional online activities. 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

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 one-relator groups, and its semigroup-theoretical 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 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) - 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 Krohn-Rhodes 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 0-simple 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 Li-Min Wang, Lu Zhang, Hai-Yuan Huang and Zhi-Yong Zhou.
  
  

• 15th July - Robert Gray (East Anglia) - Solving equations in one-relator 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 one-relator monoids.
  
  

• 22nd July - Benjamin Steinberg (City College of New York) - Homological finiteness of one-relator monoids and related monoids ( abstract hide abstract | slides )

  Abstract. The word problem for one-relator monoids is a longstanding open question. Kobayashi asked in 2000 whether every one-relator 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 one-relator monoid is of type \( FP_{\infty} \) and proved that each one-relator monoid is of type \( FP_3 \).
  
  In this talk, we discuss our proof that every one-relator 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 non-trivial 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

Past Meetings

For the archive of information about past meetings look here.

Other (non-NBSAN) 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.