mathematics

Rational monoid

Rational monoid 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 Rational monoid rather than just read about it. In short: In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can be described by a rational function. Definition Consider a monoid M.

Key takeaways

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

Reference excerpt

In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can be described by a rational function.

Definition Consider a monoid M. Consider a pair (A,L) where A is a finite subset of M that generates M as a monoid, and L is a language on A (that is, a subset of the set of all strings A∗). Let φ be the map from the free monoid A∗ to M given by evaluating a string as a product in M. We say that L is a rational cross-section if φ induces a bijection between L and M. We say that (A,L) is a rational structure for M if in addition the kernel of φ, viewed as a subset of the product monoid A∗×A∗ is a rational set. A quasi-rational monoid is one for which L is a rational relation: a rational monoid is one for which there is also a rational function cross-section of L. Since L is a subset of a free monoid, Kleene's theorem holds and a rational function is just one that can be instantiated by a finite state transducer.

Examples A finite monoid is rational. A group is a rational monoid if and only if it is finite. A finitely generated free monoid is rational. The monoid M4 generated by the set {0,e, a,b, x,y} subject to relations in which e is the identity, 0 is an absorbing element, each of a and b commutes with each of x and y and ax = bx, ay = by = bby, xx = xy = yx = yy = 0 is rational but not automatic. The Fibonacci monoid, the quotient of the free monoid on two generators {a,b}∗ by the congruence aab = bba.

Green's relations The Green's relations for a rational monoid satisfy D = J.

Properties Kleene's theorem holds for rational monoids: that is, a subset is a recognisable set if and only if it is a rational set. A rational monoid is not necessarily automatic, and vice versa. However, a rational monoid is asynchronously automatic and hyperbolic. A rational monoid is a regulator monoid and a quasi-rational monoid: each of these implies that it is a Kleene monoid, that is, a monoid in which Kleene's theorem holds.

References

Fichtner, Ina; Mathissen, Christian (2002). "Rational transformations and a Kleene theorem for power series over rational monoids". In Gomes, Gracinda M. S. (ed.). Semigroups, algorithms, automata and languages. Proceedings of workshops held at the International Centre of Mathematics, CIM, Coimbra, Portugal, May, June and July 2001. Singapore: World Scientific. pp. 94–111. Zbl 1350.68191. Hoffmann, Michael; Kuske, Dietrich; Otto, Friedrich; Thomas, Richard M. (2002). "Some relatives of automatic and hyperbolic groups". In Gomes, Gracinda M. S. (ed.). Semigroups, algorithms, automata and languages. Proceedings of workshops held at the International Centre of Mathematics, CIM, Coimbra, Portugal, May, June and July 2001. Singapore: World Scientific. pp. 379–406. Zbl 1031.20047. Kuich, Werner (2011). "Algebraic systems and pushdown automata". In Kuich, Werner (ed.). Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement. Lecture Notes in Computer Science. Vol. 7020. Berlin: Springer-Verlag. pp. 228–256. ISBN 978-3-642-24896-2. Zbl 1251.68135. Pelletier, Maryse (1990). "Boolean closure and unambiguity of rational sets". In Paterson, Michael S. (ed.). Automata, languages and programming, Proc. 17th Int. Colloq., Warwick/GB 1990. Lecture Notes in Computer Science. Vol. 443. pp. 512–525. Zbl 0765.68075. Sakarovitch, Jacques (September 1987). "Easy multiplications I. The realm of Kleene's theorem". Information and Computation. 74 (3): 173–197. doi:10.1016/0890-5401(87)90020-4. Zbl 0642.20043.

Worked examples

Example 1 — a first encounter with Rational monoid

Start with the simplest possible case. Write down what Rational monoid 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 Rational monoid 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 Rational monoid 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 Rational monoid

In research: Rational monoid 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 Rational monoid 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: Rational monoid is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic structures, Semigroup theory, so understanding it makes those chapters shorter.
In everyday life: Look for Rational monoid 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 Rational monoid in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rational monoid 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 Rational monoid out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rational monoid in simple terms?

Why does Rational monoid 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 Rational monoid?

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 Rational monoid.