chemistry

Semigroup with two elements

Semigroup with two elements is a chemistry 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 Semigroup with two elements rather than just read about it. In short: In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements: O2, the null semigroup of order two.

Key takeaways

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

Reference excerpt

In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements:

O2, the null semigroup of order two. LO2, the left zero semigroup of order two. RO2, the right zero semigroup of order two. ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only semilattice with two elements and the only non-null semigroup with zero of order two, also a monoid, and ultimately the two-element Boolean algebra; this is also isomorphic to (Z2, ·2), the multiplicative group of {0,1} modulo 2. (Z2, +2) (where Z2 = {0,1} and "+2" is "addition modulo 2"), or equivalently ({0,1}, ⊕) (where "⊕" is the logical connective "xor"), or equivalently the set {−1,1} under multiplication: the only group of order two. The semigroups LO2 and RO2 are antiisomorphic. O2, ({0,1}, ∧) and (Z2, +2) are commutative, and LO2 and RO2 are noncommutative. LO2, RO2 and ({0,1}, ∧) are bands.

Determination of semigroups with two elements Choosing the set A = { 1, 2 } as the underlying set having two elements, sixteen binary operations can be defined in A. These operations are shown in the table below. In the table, a matrix of the form

indicates a binary operation on A having the following Cayley table.

In this table:

The semigroup ({0,1}, ∧ {\displaystyle \wedge } ) denotes the two-element semigroup containing the zero element 0 and the unit element 1. The two binary operations defined by matrices in a green background are associative and pairing either with A creates a semigroup isomorphic to the semigroup ({0,1}, ∧ {\displaystyle \wedge } ). Every element is idempotent in this semigroup, so it is a band. Furthermore, it is commutative (abelian) and thus a semilattice. The order induced is a linear order, and so it is in fact a lattice and it is also a distributive and complemented lattice, i.e. it is actually the two-element Boolean algebra. The two binary operations defined by matrices in a blue background are associative and pairing either with A creates a semigroup isomorphic to the null semigroup O2 with two elements. The binary operation defined by the matrix in an orange background is associative and pairing it with A creates a semigroup. This is the left zero semigroup LO2. It is not commutative. The binary operation defined by the matrix in a purple background is associative and pairing it with A creates a semigroup. This is the right zero semigroup RO2. It is also not commutative. The two binary operations defined by matrices in a red background are associative and pairing either with A creates a semigroup isomorphic to the group (Z2, +2). The remaining eight binary operations defined by matrices in a white background are not associative and hence none of them create a semigroup when paired with A.

The two-element semigroup ({0,1}, ∧) The Cayley table for the semigroup ({0,1}, ∧ {\displaystyle \wedge } ) is given below:

This is the simplest non-trivial example of a semigroup that is not a group. This semigroup has an identity element, 1, making it a monoid. It is also commutative. It is not a group because the element 0 does not have an inverse, and is not even a cancellative semigroup because we cannot cancel the 0 in the equation 1·0 = 0·0. This semigroup arises in various contexts. For instance, if we choose 1 to be the truth value "true" and 0 to be the truth value "false" and the operation to be the logical connective "and", we obtain this semigroup in logic. It is isomorphic to the monoid {0,1} under multiplication. It is also isomorphic to the semigroup

S = { ( 1 0 0 1 ) , ( 1 0 0 0 ) } {\displaystyle S=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\0&0\end{pmatrix}}\right\}}

under matrix multiplication.

The two-element semigroup (Z2, +2) The Cayley table for the semigroup (Z2, +2) is given below:

This group is isomorphic to the cyclic group Z2 and the symmetric group S2.

Semigroups of order 3

Let A be the three-element set {1, 2, 3}. Altogether, a total of 39 = 19683 different binary operations can be defined on A. 113 of the 19683 binary operations determine 24 nonisomorphic semigroups, or 18 non-equivalent semigroups (with equivalence being isomorphism or anti-isomorphism). With the exception of the group with three elements, each of these has one (or more) of the above two-element semigroups as subsemigroups. For example, the set {−1, 0, 1} under multiplication is a semigroup of order 3, and contains both {0, 1} and {−1, 1} as subsemigroups.

Finite semigroups of higher orders Algorithms and computer programs have been developed for determining nonisomorphic finite semigroups of a given order. These have been applied to determine the nonisomorphic semigroups of small order. The number of nonisomorphic semigroups with n elements, for n a nonnegative integer, is listed under OEIS: A027851 in the On-Line Encyclopedia of Integer Sequences. OEIS: A001423 lists the number of non-equivalent semigroups, and OEIS: A023814 the number of associative binary operations, out of a total of nn2, determining a semigroup.

See also Empty semigroup Trivial semigroup (semigroup with one element) Semigroup with three elements Special classes of semigroups

References

Worked examples

Example 1 — a first encounter with Semigroup with two elements

Start with the simplest possible case. Write down what Semigroup with two elements claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In chemistry, 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 Semigroup with two elements 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 Semigroup with two elements 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 Semigroup with two elements

In research: Semigroup with two elements appears in chemistry 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 Semigroup with two elements 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: Semigroup with two elements 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 Semigroup with two elements 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 Semigroup with two elements in 20 minutes

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

Frequently asked questions

What is Semigroup with two elements in simple terms?

In mathematics, a semigroup with two elements is a semigroup for which the cardinality of the underlying set is two. There are exactly five nonisomorphic semigroups having two elements: O2, the null semigroup of order two.

Why does Semigroup with two elements matter?

Because it connects several chemistry 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 Semigroup with two elements?

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 Semigroup with two elements.