Note on $\alpha$-filters in distributive nearlattices

Abstract

In this short paper we introduce the notion of α-filter in the class of distributive nearlattices and we prove that the α-filters of a normal distributive nearlattice are strongly connected with the filters of the distributive nearlattice of the annihilators.

Quasicomplemented distributive nearlattices

“…By Lemma 1 and Theorem 3 we have R(A) = ⟨R(A), m, A⟩ is a distributive nearlattice (for more details see [5,6]).…”

“…We recall the basics results about distributive nearlattices, where our main references are [8,2,3,5,6].…”

“…Then Fi α (A) = ⟨Fi α (A), ⊔, ∩, ⇒, {1}, A⟩ is a Heyting algebra, where for each F, G ∈ Fi α (A), we have F⊔G = α(F⊻G) and F ⇒ G = α(F → G) (see [5]).…”

“…and ρ is a onto homomorphism between the distributive nearlattices A and R(A) such that Θ ⊤ = Ker(ρ) ( [5,6]). The next result give a necessary and sufficient conditions through α-filters so that ρ is 1-1.…”

“…In particular, in [3], the authors presented an alternative definition of relative annihilator in distributive nearlattices different from that given in [9] and established new equivalences of the distributivity. Later, using the results developed in [3], the notion of annihilator was studied in [4][5][6].…”