Nearlattices with an overriding operation
A nearlattice is a meet semilattice in which every initial segment is a lattice. It is said to be Boolean if all of these lattices are Boolean. Overriding of \(a\) and \(b\) is a binary operation defined as the supremum of all elements \(x\) such that \(x\) is below \(a\) and the join of \(x\) and \(b\) exists or \(x\) is below \(b\). In particular, overriding extends the partial join operation. Natural examples of overriding nearlattices are, e.g., trees and appropriate algebras of partial functions. For example, every complete semilattice is an overriding nearlattice. In a particular case, an overriding nearlattice becomes the so-called o-nearlattice where the operation of overriding coincides with join. In the paper, elementary properties of orerriding are studied, an axiomatic description is obtained and several results concerning associativity of this operation are presented. Functional o-semilattices are studied and subdirectly irreducible members of this variety are characterized. The main result is a representation theorem for Boolean nearlattices with associative overriding.
scientific article; zbMATH DE number 5872635
scientific article