science

Partial cyclic order

Partial cyclic order is a science 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 Partial cyclic order rather than just read about it. In short: In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order. Definition Over a given set, a partial cyclic order is a ternary relation R {\displaystyle R} that is: cyclic, i.e. it is invariant under a cyclic permutation: R ( x , y , z ) ⇒ R ( y , z , x ) {\displaystyle R(x,y,z)\Rightarrow R(y,z,x)} asymmetric: R ( x , y…

Key takeaways

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

Reference excerpt

In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.

Definition Over a given set, a partial cyclic order is a ternary relation R {\displaystyle R} that is:

cyclic, i.e. it is invariant under a cyclic permutation: R ( x , y , z ) ⇒ R ( y , z , x ) {\displaystyle R(x,y,z)\Rightarrow R(y,z,x)}

asymmetric: R ( x , y , z ) ⇒ R̸ ( z , y , x ) {\displaystyle R(x,y,z)\Rightarrow \not R(z,y,x)}

transitive: R ( x , y , z ) {\displaystyle R(x,y,z)} and R ( x , z , u ) ⇒ R ( x , y , u ) {\displaystyle R(x,z,u)\Rightarrow R(x,y,u)}

Constructions Various constructions of partial cyclic orders have been studied, analogous to constructions of other mathematical structures. These include the direct sum, the direct product, the power, and the Dedekind–MacNeille completion of a set.

Extensions The relationship between partial and total cyclic orders is more complex than the relationship between partial and total linear orders. To begin with, not every partial cyclic order can be extended to a total cyclic order. An example is the following relation on the first thirteen letters of the alphabet: {acd, bde, cef, dfg, egh, fha, gac, hcb} ∪ {abi, cij, bjk, ikl, jlm, kma, lab, mbc}. This relation is a partial cyclic order, but it cannot be extended with either abc or cba; either attempt would result in a contradiction. The above was a relatively mild example. One can also construct partial cyclic orders with higher-order obstructions such that, for example, any 15 triples can be added but the 16th cannot. In fact, determining whether a general ternary relation (or indeed a partial cyclic order) extends to a total cyclic order is NP-complete, since it solves 3SAT. This is in stark contrast with the problem to recognize when a binary relation extends to a linear order, which can be solved in linear time. (In particular, every poset has a linear extension, even if it is infinite, by the Szpilrajn extension theorem.) Several authors have considered the question of how complicated a non-extendable partial cyclic order must be. It is known that any partial cyclic order on 6 elements or fewer does extend to a total order. There is a single example (up to isomorphism) on 7 elements, which is constructed by adjoining an extra "point at infinity" to the standard example poset of dimension 3.

Notes

References

Worked examples

Example 1 — a first encounter with Partial cyclic order

Start with the simplest possible case. Write down what Partial cyclic order claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Partial cyclic order 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 Partial cyclic order 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 Partial cyclic order

In research: Partial cyclic order appears in science 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 Partial cyclic order 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: Partial cyclic order is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circles, Order theory, so understanding it makes those chapters shorter.
In everyday life: Look for Partial cyclic order 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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How to study Partial cyclic order in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Partial cyclic order means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Partial cyclic order out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Partial cyclic order in simple terms?

In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order. Definition Over a given set, a partial cyclic order is a ternary relation R {\displaystyle R} that is: cyclic, i.e. it is invariant under a c…

Why does Partial cyclic order matter?

Because it connects several science 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 Partial cyclic order?

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.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Partial cyclic order.