Annulets and α-ideals in a distributive lattice

Abstract

In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.

A Sheaf Representation of Distributive Pseudocomplemented Lattices

“…The kernels of *-congruences of a distributive pseudocomplemented lattice L are precisely the a-ideals of the distributive lattice, in the sense of [1]. Thus, a secondary aim of this paper is to examine the connection between a-ideals and filters.…”

“…Congruences. Throughout this paper all lattices will be distributive with 0 and 1, and our notation and terminology follow that of [1] and only if x G J implies (x]** G J. For a prime ideal P, OiP) = {x G L: x A y = 0 for some y G L\P) is an important a-ideal associated with it.…”

$0$-ideals in $0$-distributive posets

“…This fact gives the reason why some mathematicians have tried to study some types of ideals and establish their properties. Cornish [1] introduced the concept of 0-ideals in distributive lattices and obtained their properties in [2] using congruences. Jayaram [6] generalized the concept of 0-ideals in semilattices and studied their properties in [7] in the case of quasicomplemented 0-distributive semilattices.…”

Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices

“…Later, Varlet in [14] gives a similar characterization for distributive semilattices. The annihilator or annulet of an element a is the set a ı D ha; 0i D fx 2 A W x^a D 0g (see [9]). It is clear that if A is an implicative semilattice (also called relatively pseudocomplemented semilattices or Brouwerian semilattices [7]), then ha; bi is a principal ideal whose generator is the relative pseudocomplement of a with respect to b, in symbols a !…”