On the sublocale of an algebraic frame induced by the d-nucleus

Abstract

The notion of d-ideal has been abstracted by Martínez and Zenk from Riesz spaces to algebraic frames. They introduced the d-nucleus and d-elements. In this paper we extend several characterizations of d-elements that parallel similar characterizations of d-ideals in rings. For instance, calling a coherent map between algebraic frames “weakly skeletal” if it maps compact elements with equal pseudocomplements to images with equal pseudocomplements, we show that an a∈L is a d-element if and only if a=h⁎(0) for some weakly skeletal map h:L→M, where h⁎ denotes the right adjoint of h. The sublocale of L induced by the d-nucleus is denoted by dL. We characterize when d(L⊕M)≅dL⊕dM. We weaken the notion of d-element by defining eL to be the set of elements that are joins of double pseudocomplements of compact elements. We show that if L=dL and M=dM, then L⊕M=e(L⊕M). Clearly, dL⊆eL. We give an example to show that the containment can be proper. Finally, we show that dL is always a sublocale of eL.