Upper Set

Ah, another soul wading through the murky depths of abstract concepts. You want me to illuminate these mathematical pronouncements, to dress them up in something more palatable than dry academic prose. Fine. But don’t expect me to hold your hand. This is about precision, not comfort.

Subset of a Preorder Containing All Larger Elements

In the rather precise, and frankly, often tedious, world of mathematics , we sometimes encounter structures that demand a certain rigor. One such structure is a subset of a partially ordered set that possesses a particular kind of completeness. We call these “upper sets,” or more formally, upward closed sets, or even isotone sets. The name itself suggests a direction, a leaning towards the “larger” elements within the established order.

Consider a partially ordered set, let’s call it $(X, \leq)$. This means we have a set $X$ and a relation $\leq$ that tells us how elements within $X$ compare to each other. It’s “partial” because not every pair of elements needs to be comparable. But when they are, the relation behaves predictably: it’s reflexive ($a \leq a$), antisymmetric (if $a \leq b$ and $b \leq a$, then $a = b$), and transitive (if $a \leq b$ and $b \leq c$, then $a \leq c$).

Now, an “upper set,” let’s denote it $S$, is a subset of $X$ with a very specific characteristic. If you pick any element $s$ from $S$, and then you find another element $x$ in the larger set $X$ that is “larger” than $s$ (meaning $s \leq x$), then this element $x$ must also be in $S$. This is the essence of being “upward closed.” It’s like a reservoir; once something enters, anything “above” it is also contained.

The formal definition, stripped of any potential ambiguity, is this: for every $a \in S$ and every $b \in X$, if $a \leq b$, then $b \in S$. It’s a simple rule, but its implications are far-reaching.

The inverse concept, the “lower set” (or downward closed set, down set, decreasing set, initial segment, or semi-ideal), operates in the opposite direction. If $l$ is in a lower set $L$, and $x \in X$ is “smaller” than $l$ (meaning $x \leq l$), then $x$ must also be in $L$. These sets are “closed under going down.”

The terms “order ideal” or simply “ideal” are sometimes tossed around as synonyms for lower sets. However, this can be a bit misleading, especially when dealing with lattices . A lower set within a lattice isn’t necessarily a sublattice itself, which can cause some confusion if you’re not paying close attention.

Properties of Upper and Lower Sets

These sets, upper and lower, have certain predictable behaviors, like well-trained soldiers in a structured army.

• The Whole Set: The entire set $X$ itself is always an upper set. Naturally. It contains everything, so it certainly contains anything “larger” than its own elements. It’s also, trivially, a lower set.
• Intersections and Unions: If you have a collection of upper sets, their intersection is also an upper set. Think about it: if every set in the collection contains anything larger than its own elements, then the intersection, which contains elements common to all of them, must also adhere to this rule. The union of upper sets? Also an upper set. If an element is in any of the upper sets, and something is larger than it, that larger thing will also be in that same upper set, and thus in the union.
• Complements: The complement of an upper set is a lower set, and vice versa. This is a fundamental duality. If a set $S$ is “closed upwards,” its complement $X \setminus S$ must be “closed downwards.”
• Lattices of Sets: The collection of all upper sets of $X$, when ordered by inclusion , forms a complete lattice . This structure, known as the upper set lattice, is quite significant in order theory. It means that any subset of upper sets has both a least upper bound (join) and a greatest lower bound (meet) within this collection.
• Generating Sets: For any subset $Y$ of $X$, there’s a smallest upper set that contains $Y$. We denote this as $\uparrow Y$. Similarly, there’s a smallest lower set containing $Y$, denoted as $\downarrow Y$. These are called the upper and lower closures of $Y$, respectively.
• Principal Sets: A lower set is called “principal” if it’s generated by a single element, i.e., it’s of the form $\downarrow {x}$ for some $x \in X$.
• Finite Sets and Maximal Elements: For finite partially ordered sets, any lower set $Y$ can be entirely reconstructed from its maximal elements . Specifically, $Y$ is equal to the lower closure of its own maximal elements: $\downarrow Y = \downarrow \operatorname{Max}(Y)$. This is a rather elegant property.
• Order Ideals: A lower set that is also a directed set is termed an order ideal . This adds another layer of structure.
• Antichains and Upper Sets: Under certain conditions, specifically when the partial order satisfies the descending chain condition , there’s a beautiful one-to-one correspondence between antichains (sets where no two distinct elements are comparable) and upper sets. You map an antichain to its upper closure, and an upper set to its set of minimal elements. This correspondence breaks down for more general partial orders, as illustrated by the example of sets of real numbers greater than 0 and greater than 1, both mapping to the empty antichain.

Upper and Lower Closures: More Than Just Neighbors

Let’s delve deeper into these “closures.” For any element $x$ in our partially ordered set $(X, \leq)$, we have two important sets associated with it:

• The upper closure of $x$, denoted as $x^{\uparrow X}$, $x^{\uparrow}$, or $\uparrow !x$, is the set of all elements $u \in X$ such that $x \leq u$. This is precisely the smallest upper set that contains $x$.
• The lower closure of $x$, denoted as $x^{\downarrow X}$, $x^{\downarrow}$, or $\downarrow !x$, is the set of all elements $l \in X$ such that $l \leq x$. This is the smallest lower set containing $x$.

These concepts extend to arbitrary subsets. For a subset $A \subseteq X$, its upper closure is $A^{\uparrow X} = A^{\uparrow} = \bigcup_{a \in A} \uparrow !a$, and its lower closure is $A^{\downarrow X} = A^{\downarrow} = \bigcup_{a \in A} \downarrow !a$. These are the smallest upper and lower sets, respectively, that contain the entire set $A$.

These closure operations are not just arbitrary definitions; they are examples of closure operators . This means they satisfy certain axioms, similar to how topological closure works. The upper closure of a set is the intersection of all upper sets containing it, and similarly for lower sets. This is a recurring theme in mathematics: closure operators capture the idea of “smallest structure containing X” within a given context.

Ordinal Numbers: A Special Case

The concept of ordinal numbers provides a striking example of this structure. Each ordinal number is typically defined as the set of all smaller ordinal numbers. This means that if $\alpha$ is an ordinal, and $\beta$ is another ordinal such that $\alpha \subseteq \beta$ (which, for ordinals, is equivalent to $\alpha < \beta$), then $\beta$ is also considered “larger” than $\alpha$ in the ordering of set inclusion. Consequently, every ordinal number forms a lower set within the class of all ordinal numbers, which are totally ordered by set inclusion. It’s a rather elegant, if abstract, illustration of these principles.

Related Concepts

This exploration of upper and lower sets connects to a broader landscape of mathematical ideas. You’ll find related concepts in:

• Abstract simplicial complexes , also known as Independence systems , which are families of sets that are downwards-closed with respect to the containment relation.
• Cofinal sets , which are subsets $U$ of a partially ordered set $(X, \leq)$ such that for every element $x \in X$, there exists some element $y \in U$ with $x \leq y$.
• The fundamental notion of duality in order theory, where properties of upper sets often mirror those of lower sets.
• Filters and ideals in order theory, which are specific types of upper and lower sets, respectively.
• Hasse diagrams , a graphical representation of partially ordered sets that visually highlights these relationships.
• Antichains , sets where no two elements are comparable, playing a crucial role in the structure of upper sets.

It’s all about how elements relate, how they build upon each other or recede from each other. These upper and lower sets are simply formal ways of describing these inherent structures. Don’t get lost in the definitions; see the patterns.