Scattered order
Encyclopedia
In mathematics
, specifically order theory
, a scattered order is a linear order that contains no densely ordered
subset
with more than 1 element.
Hausdorff
gave the following equivalent characterization: the class of all scattered orders is the smallest class of linear orders which contains the singleton orders and is closed under well-order
ed and reverse well-ordered sums.
Laver
's theorem (generalizing Fraïssé
's conjecture) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
, a scattered order is a linear order that contains no densely ordered
Dense order
In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...
subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
with more than 1 element.
Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...
gave the following equivalent characterization: the class of all scattered orders is the smallest class of linear orders which contains the singleton orders and is closed under well-order
Well-order
In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
ed and reverse well-ordered sums.
Laver
Richard Laver
Richard Laver is an American mathematician, working in set theory. He is a professor emeritus at the Department of Mathematics of the University of Colorado at Boulder.-His main results:Among Laver's notable achievements some are the following....
's theorem (generalizing Fraïssé
Roland Fraïssé
Roland Fraïssé was a French mathematical logician. He received his doctoral degree from the University of Paris in 1953. In his thesis, Fraïssé used the back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent...
's conjecture) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order.