All Questions
1,123 questions
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Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
6
votes
1
answer
553
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Derivations of universal enveloping algebra of Lie algebras
We know a lot about derivations of Lie algebra. However, for the universal enveloping algebra of Lie algebra, we have few references about it.
My question: describing the derivations of enveloping ...
- 165
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votes
2
answers
1k
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Monoids of endomorphisms of nonisomorphic groups
Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?
- 6,962
1
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1
answer
163
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Internal commutative monoid gives commutative monad
Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object.
The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
- 2,129
4
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1
answer
134
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Joint spectral radius of $\{M,M^T\}$
Let $F$ be a bounded subset of ${\bf M}_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F_k$ the set of all products of $k$ ...
- 52.3k
3
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0
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83
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Cancellativity of a particular $2$-generated monoid presented by an infinite number of relations
Let $X = \{x, y\}$ be a two-element set, and let $H$ be the monoid defined by the presentation
$$
\langle x, y \mid x y^k x = y x y^{k+1} x y, \text{ for } k = 0, 1, 2, \ldots\rangle.
$$
That is, $H$ ...
- 10.5k
4
votes
0
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147
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Division in the universal enveloping algebra
Let $\mathfrak g$ be a (semisimple) Lie algebra, $\mathfrak b\subset \mathfrak g$ a Borel and $\mathfrak n = [\mathfrak b,\mathfrak b]$. Then I am interested in solving certain division problems in $U(...
- 241
3
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40
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Closest generators for matrix algebra which is not semisimple
Given a collection of $n$ commuting $n \times n$ matrices $A_1, \dots, A_n \subset M_n (\mathbb{R})$ which generate a semisimple algebra $\mathcal{A}$, I am interested in finding matrices $E_1, \dots, ...
- 131
2
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0
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60
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Are there finitely-presented astral monoids?
We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that
whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
- 6,652
7
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260
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Generating the monoid of injective endomorphisms of the free group
Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...
- 121
6
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1
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265
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Testing ideal membership in the Weyl algebra: a simple example
In Example 1.1.4 of the book Grobner Deformations of Hypergeometric Differential Equations, it is stated without proof that
$$\partial^2 \in D\cdot \langle x\partial^4, x^3\partial^2 \rangle \tag{$\...
- 215
2
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1
answer
153
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Define a homomorphism of a set of graphs to its power set
Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is,
$G_1\cup G_2$
$=\langle V(G_1)\cup V(G_2), (E(G_1)\...
- 203
0
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1
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296
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Multiplicative monoid of ring modulo units
Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\...
- 943
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103
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Lattices with trivial coinvariants for finite groups
Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...
- 2,160
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132
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Generalization of pseudogroups
Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
- 181
8
votes
2
answers
585
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Is the equational theory of groups axiomatized by the associative law?
Consider the class of groups in the signature {*}. Is the equational theory of that class axiomatized by the associative law? I asked this on math stack exchange but I didn't receive a satisfactory ...
- 2,023
10
votes
3
answers
1k
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Dual of a bimodule
For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module.
Note: ...
5
votes
1
answer
303
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A characterisation of faces of rational polyhedral cones
This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
- 6,700
7
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579
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Guises of the noncrossing partitions (NCPs)
From "Noncrossing partitions in surprising locations" by Jon McCammond:
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
- 10.5k
5
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1
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170
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Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?
McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents.
Question: Which homotopy types ...
- 63.9k
1
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0
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214
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Shape of possible counterexamples to the Jacobian and Dixmier Conjectures
Let $k$ be a field of characteristic zero.
It is well-known, see for example Corollary 10.2.21, that if $(x,y) \mapsto (p,q) \in k[x,y]^2$ is a counterexample to the two-dimensional Jacobian ...
- 2,837
10
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Who invented Monoid?
I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...
- 203
3
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0
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72
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Splitting of central simple algebras in the Schur subgroup over residue fields of places
Recall that a valuation domain of a field extension $K/k$ is a $k$-subalgebra $V$ of $K$ not equal to $K$ such that for every $a\in K$ at least one of $a$ and $a^{-1}$ is in $V$.
A place of $K/k$...
2
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42
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Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...
- 2,837
3
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93
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Is a specific endomorphism of $A_1$ an automorphism?
Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
...
- 2,837
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Partially commutative elements in powers of augmentation ideal
Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
- 121
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200
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A non-commutative analog of a result concerning a Jacobian pair
Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...
- 2,837
7
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1
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207
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Relative Dickson (trace) criterion for Jacobson radical?
In the following, all algebras are associative and unital. Let $J\left(A\right)$ denote the Jacobson radical of an arbitrary algebra $A$. Recall that this is defined as the set of all $a \in A$ such ...
- 33.8k
5
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1
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205
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Topological category of topological monoids / operads
The category of topological monoids can be made into a topological category in a naive way. Namely, the space of all continuous homomorphisms between two topological monoids is a closed subspace of ...
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4
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1
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307
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Characterization of Archimedean linearly ordered monoids
In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...
- 249
2
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1
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310
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Flatness of submodules of free modules
Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
- 5,901
6
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0
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47
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Special monomorphism to encode the inclusion of topological submonoids
Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms.
Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...
- 2,129
1
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0
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69
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How exactly to adapt Brown's collapse from monoids to algebras?
In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly:
Given a simplicial set $X$ equipped with a ...
0
votes
1
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84
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Primage structures: induced domain partitioning by itterated inverse (reference request)
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example, the j-th such preimage list ...
- 23
4
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1
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368
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Possible values of symmetric functions evaluated on quaternions
$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be the units of quaternions, in particular $i^2=j^2=k^2=-1$, $ijk=-1$.
We will use non commutative variables $x$, $y$, $z$. Define $\sym_{a,b,c}$ to be ...
- 53
12
votes
3
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849
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Subalgebra of a group algebra
Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra.
Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$.
Question: Is there any ...
- 5,230
2
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1
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195
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A question about semigroup union
The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T_n$ is denoted $C_n$.
I consider the idempotent set $A=\{\begin{bmatrix}2\\1 \end{bmatrix},\...
- 355
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1
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Origin of the relations of Leavitt path algebras
I know the formal definition of Leavitt path algebras, but I want know why the relations defining Leavitt Path Algebras are defined in that way? what is special of this relations?
My real hidden ...
- 51
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0
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202
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What is the normalized complex of a simplicial set with a monoid action?
This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though.
In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...
5
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3
answers
2k
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Ideal structure of a tensor product of certain algebras
I would be grateful if anyone could give me a reference regarding the following question.
Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
- 423
3
votes
1
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244
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Category of continuous self maps
Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous ...
- 2,772
1
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0
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132
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Is the Upper Banach density always zero with respect to some sequence of Finite subset
The following question came to me while reading the paper 'Density in Arbitrary Semigroups' by Hindman and Strauss.
Question: Given an infinite subset $A$ of $\mathbb{N}$ such that $A^c$ is also ...
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3
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1
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62
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Is a J-simple semigroup with an idempotent necessarily regular?
If a semigroup S has no proper ideals can it have both regular and non-regular members? My guess would be 'yes' but in that case does anyone know of an example in the literature?
- 313
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Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
- 1,528
6
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1
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214
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Weak associativity
Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...
1
vote
0
answers
55
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Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
-3
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234
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A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
- 41.9k
1
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0
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111
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When a semigroup ideal is a determinantal ideal?
Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
- 113
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1
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254
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Examples of Yang-Baxter monoids
Then we say that an algebra $(X,f,g,\circ,1)$ is a Yang-Baxter monoid if it satisfies the following identities:
$(X,\circ,1)$ is a monoid,
$f(x,1)=1,f(1,x)=x,g(x,1)=x,g(1,x)=1$
$x\circ y=f(x,y)\circ ...
2
votes
1
answer
229
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Has the "semidirect monoid of a semiring" been considered anywhere?
Given a semiring $S$, we get a monoid $M(S)$ as follows:
The underlying set of $S$ is $S^2$
The identity element is $(0,1)$
The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
- 3,793