The dual symmetric inverse monoid I ∗ n is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of I ∗ n and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

A subsemigroup S of an inverse semigroup Q is a left I-order in Q, if every element in Q can be written as a−1b where a, b ∈ S and a−1 is the inverse of a in the sense of inverse semigroup theory. We study a characterisation of semigroups which have a primitive inverse semigroup of left I-quotients.

The dual symmetric inverse monoid I ∗ n is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of I ∗ n and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

A Tychonoff space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space. It is shown that this condition is equivalent to the Dedekind– MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi. A compact metric space is a uniform quotients space precisely when it has a dense set of isolated points. Extremally dis...

As my contribution to these proceedings, I will discuss the geometric invariant theory quotients of toric varieties. Specifically, I will show that quotients of the same problem with respect to different linearizations are typically related by a sequence of flips (or more precisely, log flips) in the sense of Mori, which in the quasi-smooth case can be characterized quite precisely. This seems ...

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions o...

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series ∏ 1≤n6≡0,±i (mod 2k+1) 1 1− qn . This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of (2, 2k + 1) Virasoro minimal models. We determine the vanishing of such W...

In the first part of this talk (two weeks ago), I sketched my work on the lattice theory of the weak order. The key point was that the lattice structure provides an alternate algebraic structure on a finite Coxeter group. One interesting aspect of this alternate structure is the study of quotients of the weak order modulo lattice congruences. The talk will start with a very brief recap of the r...