Inspired by the ring theory concepts of orders and classical rings of quotients, Fountain and Petrich introduced the notion of a completely 0-simple semigroup of quotients in [19]. This was generalised to a much wider class of semigroups by Gould in [20]. The notion extends the well known concept of group of quotients [8]. To give the definition we first have to explain what is meant by a squar...

For integers k ≥ 2, we consider the integrality of quotients of Wronskians involving certain normalizations of the Andrews-Gordon q-series (see [2]) ∏ 1≤n "≡ 0,±i (mod 2k+1) 1 1 − qn . This study is motivated by the appearance of these series in conformal field theory as irreducible characters of Virasoro minimal modules. We establish an upper bound on primes p for which such quotients of Wrons...

Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing for every even degree d algebraic numbers of degree d that have bounded complex partial quotients in their Hu...

A central principle of this paper is that, for a finitely presented group G, the algebraic properties of its finite index subgroups should be reflected by the geometry of its finite quotients. These quotients can indeed be viewed as geometric objects, in the following way. If we pick a finite set of generators for G, these map to a generating set for any finite quotient and hence endow this quo...

A right ideal is a language L over an alphabet Σ that satisfies L = LΣ∗. We show that there exists a stream (sequence) (Rn | n > 3) of regular right ideal languages, where Rn has n left quotients and is most complex under the following measures of complexity: the state complexities of the left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the ...

In this paper, we study inverses and quotients of mappings between ordered sets, in particular between complete lattices, which are analogous to inverses and quotients of positive numbers. We investigate to what extent a generalized inverse can serve as a left inverse and as a right inverse, and how an inverse of an inverse relates to the identity mapping. The generalized inverses and quotients...

We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application ...

We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space M0,n of stable genus zero n-pointed cu...