Onion lattices and an answer to an open problem on convolution lattices

Abstract

We develop a new class of lattices called Onion lattices, analyse the properties of Onion lattices and further prove that the set of idempotent lattice functions on a bounded lattice L is closed under convolution operations if and only if L is an Onion lattice. This solves the second open problem posed by De Miguel, Bustince and De Baets (2018) [1]. Furthermore, we study in depth the lattice function algebraic system with the properties of the Birkhoff system proposed in [1]. We obtain that the set of idempotent lattice functions under the operations ⊓ and ⊔ constitutes a Birkhoff system if and only if the domain of the idempotent lattice functions is the Onion lattice. And we explore some algebraic properties of Birkhoff systems consisting of sets of idempotent lattice functions.