The paper introduces a definition of symmetric difference in lattices with negation, presents its general properties and studies those that are typical of ortholattices, orthomodular lattices, De Morgan and Boolean algebras.
-continuous posets are common generalizations of continuous posets, completely distributive lattices, and unique factorization posets. Though the algebraic properties of -continuous posets had been studied by several authors, the topological properties are rather unknown. In this short note an intrinsic topology on a -continuous poset is defined and its properties are explored.
All ordinal numbers with the following property are found: there exists a loop such that its subloops form a chain of ordinal type .
It is consistent that there is a partial order (P,≤) of size such that every monotone function f:P → P is first order definable in (P,≤).
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This...