11,416 research outputs found
The multiplicative semigroup of a Dedekind domain
In 1995 Grillet defined the concept of a stratified semigroup and a
stratified semigroup with zero. The present authors extended that idea to
include semigroups with a more general base and proved, amongst other things,
that finite semigroups in which the H-classes contain idempotents, are
semilattices of stratified extensions of completely simple semigroups, and
every strict stratified extension of a Clifford semigroup is a semilattice of
stratified extensions of groups. We continue this work here by considering the
multiplicative semigroup of Dedekind domains and show in particular that
quotients of such rings have a multiplicative structure that is a (finite)
Boolean algebra of stratified extensions of groups
The square root law and structure of finite rings
Let be a finite ring and define the hyperbola . Suppose that for a sequence of finite odd order rings of size tending
to infinity, the following "square root law" bound holds with a constant
for all non-trivial characters on : Then, with a finite number of
exceptions, those rings are fields.
For rings of even order we show that there are other infinite families given
by Boolean rings and Boolean twists which satisfy this square-root law
behavior. We classify the extremal rings, those for which the left hand side of
the expression above satisfies the worst possible estimate. We also describe
applications of our results to problems in graph theory and geometric
combinatorics.
These results provide a quantitative connection between the square root law
in number theory, Salem sets, Kloosterman sums, geometric combinatorics, and
the arithmetic structure of the underlying rings
On EQ-monoids
An EQ-monoid A is a monoid with distinguished subsemilattice L with 1 2 L and such that any a, b 2 A have a largest right equalizer in L. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQmonoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids
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