mathematics

Near-semiring

Near-semiring 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 Near-semiring rather than just read about it. In short: In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.

Key takeaways

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

Reference excerpt

In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.

Definition A near-semiring is a set S with two binary operations "+" and "·", and a constant 0 such that (S, +, 0) is a monoid (not necessarily commutative), (S, ·) is a semigroup, these structures are related by a single (right or left) distributive law, and accordingly 0 is a one-sided (right or left, respectively) absorbing element. Formally, an algebraic structure (S, +, ·, 0) is said to be a near-semiring if it satisfies the following axioms:

(S, +, 0) is a monoid, (S, ·) is a semigroup, (a + b) · c = a · c + b · c, for all a, b, c in S, and 0 · a = 0 for all a in S. Near-semirings are a common abstraction of semirings and near-rings [Golan, 1999; Pilz, 1983]. The standard examples of near-semirings are typically of the form M(Г), the set of all mappings on a monoid (Г; +, 0), equipped with composition of mappings, pointwise addition of mappings, and the zero function. Subsets of M(Г) closed under the operations provide further examples of near-semirings. Another example is the ordinals under the usual operations of ordinal arithmetic (here Clause 3 should be replaced with its symmetric form c · (a + b) = c · a + c · b. Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a class near-semiring. We get a near-semiring in the standard sense if we restrict to those ordinals strictly less than some multiplicatively indecomposable ordinal.

Bibliography Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR 1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR 1746739 Krishna, K. V., Near-semirings: Theory and application, Ph.D. thesis, IIT Delhi, New Delhi, India, 2005. Pilz, G., Near-Rings: The Theory and Its Applications, Vol. 23 of North-Holland Mathematics Studies, North-Holland Publishing Company, 1983. The Near Ring Main Page at the Johannes Kepler Universität Linz Willy G. van Hoorn and B. van Rootselaar, Fundamental notions in the theory of seminearrings, Compositio Mathematica v. 18, (1967), pp. 65–78.

Worked examples

Example 1 — a first encounter with Near-semiring

Start with the simplest possible case. Write down what Near-semiring 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 Near-semiring 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 Near-semiring 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 Near-semiring

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

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

Frequently asked questions

What is Near-semiring in simple terms?

In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.

Why does Near-semiring 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 Near-semiring?

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 Near-semiring.