A partial monoid P is a set with a partial multiplication × (and total identity 1P ) which satisfies some associativity axiom. The partial monoid P may be embedded in the free monoid P ∗ and the product × is simulated by a string rewriting system on P ∗ that consists in evaluating the concatenation of two letters as a product in P , when it is defined, and a letter 1P as the empty word ǫ. In th...

Given a finitely presented monoid and a homotopy base for the monoid, and given an arbitrary Schützenberger group of the monoid, the main result of this paper gives a homotopy base, and presentation, for the Schützenberger group. In the case that the R-class R of the Schützenberger group G (H) has only finitely many H -classes, and there is an element s of the multiplicative right pointwise sta...

We demonstrate that categories of continuous actions topological monoids on discrete spaces are Grothendieck toposes. exhibit properties these toposes, giving a solution to the corresponding Morita-equivalence problem. characterize toposes in terms their canonical points. identify natural classes representatives with good properties, 'powder monoids' and then 'complete monoids', for monoids. Fi...

We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion o...

Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid CM(Q), as introduced by M. Reineke, and the generic composition algebra Cq(Q), as introduced by C. M. Ringel, specialised at q = 0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then C0(Q) is isomorphic to the monoid algebra QCM(Q). Moreover,...

for their help and suggestions. Special thanks go to J. Araujo for his mathematical suggestions. I would also like to acknowledge the support of the Centre of Algebra at the University of Lisbon, and of EPSRC grant number GR/S/56085/01. Colophon If you use the MONOID package, I would really appreciate it if you would let me know by sending me an email to [email protected]. If you notice that th...

We study the problem of testing whether a context-free language is included in a fixed set L0, where L0 is the language of words reducing to the empty word in the monoid defined by a complete string rewrite system. We prove that, if the monoid is cancellative, then our inclusion problem is polynomially reducible to the problem of testing equivalence of straight-line programs in the same monoid....