mathematics

Glossary of order theory

Glossary of order theory is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Glossary of order theory rather than just read about it. In short: This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well.

Key takeaways

Glossary of order theory belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Glossary of order theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Glossary of order theory from memory before moving on to harder problems.

Reference excerpt

This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:

completeness properties of partial orders distributivity laws of order theory In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, ≤ {\displaystyle \,\leq \,} will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by ≤ . {\displaystyle \,\leq .}

A Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric. Adjoint. See Galois connection. Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open. Algebraic poset. A poset is algebraic if it has a base of compact elements. Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation. Approximates relation. See way-below relation. Antisymmetric relation. A homogeneous relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for all elements x, y in X. Antitone. An antitone function f between posets P and Q is a function for which, for all elements x, y of P, x ≤ y (in P) implies f(y) ≤ f(x) (in Q). Another name for this property is order-reversing. In analysis, in the presence of total orders, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called monotone or order-preserving. Asymmetric relation. A homogeneous relation R on a set X is asymmetric, if x R y implies not y R x, for all elements x, y in X. Atom. An atom in a poset P with least element 0, is an element that is minimal among all elements that are unequal to 0. Atomic. An atomic poset P with least element 0 is one in which, for every non-zero element x of P, there is an atom a of P with a ≤ x.

B Base. See continuous poset. Binary relation. A binary relation over two sets X and Y {\displaystyle X{\text{ and }}Y} is a subset of their Cartesian product X × Y . {\displaystyle X\times Y.}

Boolean algebra. A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ∧ ¬x = 0 and x ∨ ¬x = 1. Bounded poset. A bounded poset is one that has a least element and a greatest element. Bounded complete. A poset is bounded complete if every of its subsets with some upper bound also has a least such upper bound. The dual notion is not common.

C Chain. A chain is a totally ordered set or a totally ordered subset of a poset. See also total order. Chain complete. A partially ordered set in which every chain has a least upper bound. Closure operator. A closure operator on the poset P is a function C : P → P that is monotone, idempotent, and satisfies C(x) ≥ x for all x in P. Compact. An element x of a poset is compact if it is way below itself, i.e. x<<x. One also says that such an x is finite. Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x. Comparability graph. The comparability graph of a poset (P, ≤) is the graph with vertex set P in which the edges are those pairs of distinct elements of P that are comparable under ≤ (and, in particular, under its reflexive reduction <). Complete Boolean algebra. A Boolean algebra that is a complete lattice. Complete Heyting algebra. A Heyting algebra that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts frame and locale. Complete lattice. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist. Complete partial order. A complete partial order, or cpo, is a directed complete partial order (q.v.) with least element. Complete relation. Synonym for Connected relation. Complete semilattice. The notion of a complete semilattice is defined in different ways. As explained in the article on completeness (order theory), any poset for which either all suprema or all infima exist is already a complete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of a complete lattice. In other cases, complete (meet-) semilattices are defined to be bounded complete cpos, which is arguably the most complete class of posets that are not already complete lattices. Completely distributive lattice. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets. Completion. A completion of a poset is an order-embedding of the poset in a complete lattice. Completion by cuts. Synonym of Dedekind–MacNeille completion. Connected relation. A total or complete relation R on a set X has the property that for all elements x, y of X, at least one of x R y or y R x holds. Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every element x of P is the supremum of a directed set contained in {y in B | y<<x}. Continuous function. See Scott-continuous. Converse. The converse <° of an order < is that in which x <° y whenever y < x. Cover. An element y of a poset P is said to cover an element x of P (and is called a cover of x) if x < y and there is no element z of P such that x < z < y. cpo. See complete partial order.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Glossary of order theory

Start with the simplest possible case. Write down what Glossary of order theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Glossary of order theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Glossary of order theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Glossary of order theory

In research: Glossary of order theory appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Glossary of order theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Glossary of order theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Glossaries of mathematics, Order theory, so understanding it makes those chapters shorter.
In everyday life: Look for Glossary of order theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Glossary of order theory in simple terms?

Why does Glossary of order theory matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Glossary of order theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

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