Abstract
The purpose of this paper is to study annihilators and annihilator ideals in a more general context; in universal algebras.
Highlights
Annihilators and annihilator ideals have been studied in different classes of algebras
The concept of annihilators in lattices was studied by Mandelker [17] and later extended to the class of distributive lattices by Cornish [18], Speed [19], and Davey [20]
In [21], Davey and Nieminen studied the structure of annihilators in modular lattices
Summary
Annihilators and annihilator ideals have been studied in different classes of algebras. Ey have defined and characterized the commutator (or the product) of ideals in universal algebras. Ursini [30] applied this product to define and study prime ideals in universal algebras. He has defined relative annihilator of ideals under a name “residual of ideals.”. By applying the concept of relative annihilators, it is shown that the class of ideals of an algebra A with a c-unit forms a complete residuated lattice. We obtain a class of algebras called c-idempotent algebras, for which the lattice of all annihilator ideals of each algebra forms a Boolean algebra. It is evident that A is idempotent if and only if every singleton set {x} in A is a subuniverse of A
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