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

Past Meetings

For the archive of information about past meetings look here.