The linear transformation that relates the canonical and coefficient embeddings of ideals in cyclotomic integer rings

The geometric embedding of an ideal in the algebraic integer ring of some number field is called an ideal lattice. Ideal lattices and the shortest vector problem (SVP) are at the core of many recent developments in lattice-based cryptography. We utilize the matrix of the linear transformation that relates two commonly used geometric embeddings to provide novel results concerning the equivalence of the SVP in these ideal lattices arising from rings of cyclotomic integers.

THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

COMPLETE INTERSECTIONS IN BINOMIAL AND LATTICE IDEALS

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set-theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set-theoretic complete intersection is a complete intersection.

Ideal: A Software Tool to Evaluate Interface Usability

This paper reports on the design, prototype implementation, and formative evaluation of a software tool—IDEAL (Interface Design Environment and Analysis Lattice). IDEAL integrates usability engineering techniques and behavioral task representations with a graphical hierarchy of user tasks to support formative evaluation of an evolving user interface. Representative users of IDEAL—interface designers and evaluators—participated in two phases of formative evaluation of IDEAL. Empirical evaluation showed IDEAL to be useful as an automated tool for managing the interrelated tasks of user interface development, including interaction design, usability specification, creation of benchmark tasks, and formative evaluation, that are currently performed manually.

Discrete structure spaces of ƒ-rings

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.