Title: Theory / Unimportant / Math / Al gebra / Semigroup theory / Howie+10
Tags: fromImplication defToAlgebra fromMath setMultiplication toBinaryRelation hasSeenConjDef récupéré toBinaryFunction closed examplesSwitchExample1 mathToSee setAdditionMultiplication EquivalentButNoIntermediate addIntermediateCard exampleToQuestion mid1516974248815 toUnary
On every note of this deck, unless a note says otherwise:
Chapter: 1

Name: A groupoid
Definition: A pair \((S,\mu)\) with \(S\) a set and \(\mu:S\times S\to S\)
Page: 1, wp
Name2: magma

Name: A semigroup
Definition: A groupoid \((S,\mu)\) such that \(\mu\) is associative
Page: 1

Name: An associative operation on \(S\)
Definition: An operation \(\mu:S\times S\to S\) such that \(((x,y)\mu,z)\mu=(x,(y,z)\mu)\mu\) for each \(x,y,z\in S\).
Page: 1
Variables: \(S\) a set

Definition: \((\dots(x_1x_2)\dots) x_n\)
Page: 1
Variables: \((S,\cdot)\) a semigroup,
\(x_1,\dots,x_n\in S\)
Notation: \(x_1\dots x_n\)
Definition2: \(x_1(\dots (x_{n-1}x_n)\dots)\)

Definition: -\(x\) if \(n=1\) and
-\(x^{n-1}x\) if \(n>1\)
Page: 1
Variables: \((S,\cdot)\) a semigroup,
\(x\in S\),
\(n\in\mathbb N^{>0}\)
Notation: \(x^n\)

Name: A commutative operation on \(S\)
Definition: An operation \(\cdot\) such that \(xy=yx\) for each \(x,y\in S\).
Page: 1, 2
Variables: \(S\) a set,
Name2: An Abelian operation on \(S\)

Name: An identity element
Definition: An element \(e\) such that \(xe=ex=x\) for each \(x\in X\)
Page: 2
Variables: \((S,\cdot)\) a semigroup

Name: A monoid
Definition: A semigroup with identity
Page: 2

Question: Its number of identity is
Answer: at most one.
Page: 2
Chapter: 1
Variables: \((S,\cdot)\) a semigroup

Page: 2
Chapter: 1
Name: The monoid obtained from \(S\) by adjoining a zero if necessary
Set: \(S\) if \(S\) has a zero element
\(S\cup\{0\}\)
Variables: \((S,\cdot)\) a semigroup
Notation: \(S^0\)
*: \((x_0,x_1)\mapsto \left\{\begin{array}{ll}0&\text{if }x_{i}=0\\x_0x_1&\text{otherwise}\end{array}\right.\)
Units: Units of \(S\)
Zero divisors: Zero divisors of \(S\) if it had a zero

Name: A zero element
Definition: An element \(e\in S\) such that \(ex=xe=e\) for each \(x\in S\)
Page: 2
Variables: \((S,\cdot)\) a semigroup
Notation: 0
Cardinal: At most one

Page: 2
Chapter: 1
Name: The trivial semigroup
Set: \(\{e\}\)
*: Trivial
Units: \(e\)
Zero divisors: none
Commutative: y
Relation: \(ee=e\)

Page: 2
Chapter: 1
Name: The monoid obtained from \(S\) by adjoining an identity if necessary
Set: \(S\) if \(S\) has an identity element
\(S\cup\{1\}\) otherwise
Variables: \((S,\cdot)\) a semigroup
Notation: \(S^1\)
*: \((x_0,x_1)\mapsto \left\{\begin{array}{ll}x_{i}&\text{if }x_{1-i}=1\\x_0x_1&\text{otherwise}\end{array}\right.\)
Units: Units of \(S\) if it had one
Zero divisors: Zero of \(S\)

Name: A null semigroup
Definition: A semigroup of the form \((S,(x,y)\mapsto 0)\)
Page: 3

Page: 3
Chapter: 1
Name: A left-zero semigroup
Set: An arbitrary set semigroup
*: \((x,y)\mapsto x\)

Name: Semigroup with 0 and 1
Page: 3
Example: \(([0,1],\min)\)
Example2: \(\mathbb N, \times\)

Name: \(AB\)
Definition: \(\{ab\mid a\in A, b\in B\}\)
Page: 3
Variables: \((S,\cdot)\) a semigroup,
\(A,B\subseteq S\)

Name: \(Ab\)
Definition: \(A\{b\}\)
Page: 3
Variables: \((S,\cdot)\) a semigroup,
\(A\subseteq S\),
\(b\in S\)

Name: \(aS^1\)
Definition: \(aS \cup\{a\}\)
Page: 3
Variables: \((S,\cdot)\) a semigroup,
\(a\in S\)

Name: \(S^1a\)
Definition: \(Sa \cup\{a\}\)
Page: 3
Variables: \((S,\cdot)\) a semigroup,
\(a\in S\)

Name: \(S^1aS^1\)
Definition: \(Sa \cup\{a\}\cup aS\cup SaS\)
Page: 3
Variables: \((S,\cdot)\) a semigroup,
\(a\in S\)

Name: A group
Definition: A semigroup such that \(aS=S\) and \(Sa=S\) for each \(a\in S\)
Page: 3
Context: Semigroup theory

Name: A group-with-zero
Definition: \(G^0\) for \(G\) a group
Page: 4
Name2: A 0-group

Smaller0: (\(aS=S\) and \(Sa=S\)) for each \(a\in S\setminus \{0\}\)
Greater0: \(S\) is a \(0\)-group
Page: 4
Chapter: 1
Variables: \((S,\cdot)\) a semigroup with 0
Smaller to greater: iff

Name: A subsemigroup
Definition: A set \(T\subseteq S\) such that \(T^2\subseteq T\)
Page: 4
Variables: \((S,\cdot)\) a semigroup
Definition2: A non-empty subset of \(S\) closed under \(\cdot\)
Example: \(\{e\}\) for \(e\) idempotent

Name: An idempotent
Definition: An element \(e\in S\) such that \(e^2=e\).
Page: 4
Variables: \((S,\cdot)\) a semigroup

Name: A subgroup of \(S\)
Definition: A subsemigroup which is a group
Page: 4
Variables: \((S,\cdot)\) a semigroup
Example: \(\{e\}\) for \(e\in S\) idempotent
\(\{x^n\mid n\in \mathbb N\}\) for \(x\in S\)

Name: An ideal
Definition: A left and right ideal
Page: 4
Variables: \((S,\cdot)\) a semigroup
Example: \(S\),
Example2: \(\{0\}\) if \(S\) admits a 0

Smaller0: \(A\) is an ideal
Greater0: \(A\) is a subsemigroup
Page: 4
Chapter: 1
Variables: \((S,\cdot)\) a semigroup,
\(A\subseteq S\)
Counterexample: -\(2\mathbb N\) is a subsemigroup of \((\mathbb N,+)\) but not an ideal
Intermediate: \(A\) is a left (or right) ideal
Counterexample2: -trivial semigroup

Name: A proper ideal
Definition: An ideal which is neither \(S\) nor the zero singleton
Page: 5
Variables: \((S,\cdot)\) a semigroup

Name: A morphism of semigroup
Definition: Given \((S,\cdot)\), \((T,\cdot)\) semigroups, a function \(\phi:S\to T\) such that \((xy)\phi=(x\phi)(y\phi)\) for each \(x,y\in S\)
Page: 5
Name2: An homomorphism of semigroup

Name: A morphism of monoid
Definition: A morphism of semigroup, between monoid, sending identity to identity.
Page: 5
Name2: An homomorphism of monoid

Name: The domain of \(\phi\)
Definition: \(S\)
Page: 5
Variables: \(\phi:S\to T\) a morphism

Name: The codomain of \(\phi\)
Definition: \(T\)
Page: 5
Variables: \(\phi:S\to T\) a morphism

Name: The image of \(\phi\)
Definition: \(\{x\phi \mid x\in S\}\)
Page: 5
Variables: \(\phi:S\to T\) a morphism
Name2: The range of \(\phi\)

Name: A monomorphism
Definition: A one to one morphism \(\phi:S\to T\),
Page: 5
Definition2: \(\alpha\phi=\beta\phi\) implies \(\alpha=\beta\) for each morphism \(\alpha,\beta:T\to U\) for each \(U\)

Name: An isomorphism
Definition: A bijective morphism
Page: 5
Definition2: A left and right invertible morphism

Page: 5
Definition: Two semigroup such that there exists an isomorphism from one to the other
Chapter: 1
Name: Isomorphic semigroup

Name: An endomorphism
Definition: A morphism whose domain and codomains are equal
Page: 5

Name: An automorphism
Definition: An isomorphic endomorphism
Page: 5

Page: 5
Chapter: 1
Name: The direct product of the \(S_i\)'s
Set: domain: functions from \(I\) to \(\bigcup_{i\in I} S_i\), sending \(i\) to an element of \(S_i\),
Variables: \(I\) an indexing family,
\(((S_i,\cdot_i))_{i\in I}\) a set of semigroups
Notation: \(\Pi_{i\in I}S_i\)
*: component-wise multiplication
Units: Product of units
Zero divisors: Products of zero divisors or elements containing one zero

Name: The projection morphism
Definition: [$]\defFun{\Pi_{j\in I}S_j}{S_i}{p}{ip}[/$]
Page: 6
Variables: \(I\) an indexing family,
let \(((S_i,\cdot_i))_{i\in I}\) a set of semigroups,
\(i\in I\)
Notation: \(\pi_i\)
Extra: recall that \(p:I\to\bigcup S_i\) and functions are written on the right of their argument

Name: The full transformation semigroup of \(X\)
Definition: its composition law
Page: 6
Variables: \(X\) a set
Notation: \(\mathcal T_X\)
Definition2: the set of functions with domain and codomain \(X\),
Conjdef: The semigroup composed of

Page: 6
Chapter: 1
Name: The symetric semigroup of \(X\)
Set: the set of bijections with domain and codomain \(X\)
Variables: \(X\) a set
Notation: \(\mathcal G_X\)
*: composition law
1: identity
÷: bijection's inverse
Typ: Subsemigroup of \(\mathcal T_X\)

Question: The cardinality of \(\mathcal G_X\)
Answer: \(n!\) with \(n\) being \(X\)'s cardinality
Page: 6
Chapter: 1
Variables: \(X\) a set

Question: The cardinality of \(\mathcal T_X\)
Answer: \(n^n\) with \(n\) being \(X\)'s cardinality
Page: 6
Chapter: 1
Variables: \(X\) a set

Name: The extended right regular representation of \(S\)
Definition: The map [$]\defFun{S}{\mathcal S^1}{a}{(x\mapsto xa)}[/$]
Page: 7
Variables: \(S\) a semigroup

Name: The extended right regular representation of \(S\)
Definition: The map [$]\defFun{S}{\mathcal S^1}{a}{(x\mapsto ax)}[/$]
Page: 7
Variables: \(S\) a semigroup

Name: A band
Definition: A semigroup whose elements are idempotent
Page: 7
Context: Semigroup
Index: 8

Name: Subsemigroup
Page: 8
Section: 2
Chapter: 1,
Under: non-empty intersections

Definition: The least submonoid containing \(A\)
Page: 9
Section: 2
Variables: \(S\) a monoid,
\(A\subseteq S\)
Notation: \(\langle A\rangle\)

Name: The subsemigroup generated by \(a\)
Definition: \(\{a^i\mid i\in\mathbb N^{>0}\}\)
Page: 9
Section: 2
Variables: \(S\) a semigroup,
\(a\in S\)
Notation: \(\langle a\rangle\)
Name2: The monogenic semigroup generated by \(a\)

Name: The submonoid generated by \(a\)
Definition: \(\{a^i\mid i\in\mathbb N\}\)
Page: 9
Section: 2
Variables: \(S\) a monoid,
\(a\in S\)
Notation: \(\langle a\rangle\)
Name2: The monogenic monoid generated by \(a\)

Definition: The least subsemigroup containing \(A\)
Page: 9
Section: 2
Variables: \(S\) a semigroup,
\(A\subseteq S\)
Notation: \(\langle A\rangle\)

Name: A monogenic semigroup
Definition: A semigroup \(S\) such that there exists \(a\in S\) such that each element is a power of \(a\)
Page: 9
Section: 2

Name: An element of infinite order
Definition: An element \(a\in S\) such that \(a^m=a^n\) implies \(m=n\)
Page: 9
Section: 2
Variables: \(S\) a semigroup

Name: An infinite monogenic semigroup
Definition: A semigroup equivalent to \((\mathbb N,+)\)
Page: 9
Section: 2

Name: The index of \(a\)
Definition: The least \(m\) such that \(a^{m+x}=a^m\) for some \(x>0\).
Page: 10
Section: 2
Variables: \(S\) a finite monogenic semigroup,
\(a\in S\)

Name: The period of \(a\)
Definition: The least \(i\) such that \(a^{m+i}=a^m\) for some \(i>0\).
Page: 10
Section: 2
Variables: \(S\) a finite monogenic semigroup,
\(a\in S\),
\(m\) its period

Name: The index of \(S\)
Definition: The index of one of its generator
Page: 10
Section: 2
Variables: \(S\) a finite monogenic semigroup

Name: The period of \(S\)
Definition: The period of one of its generator
Page: 10
Section: 2
Variables: \(S\) a finite monogenic semigroup,

Name: A generator of \(S\)
Definition: An element \(a\in S\) such that each element is a power of \(a\)
Page: 10
Section: 2
Variables: \(S\) a semigroup
Assuming: \(S\) is monogenic

Name: The kernel of \(\langle a\rangle\)
Definition: The set \(\{a^m,\dots,a^{m+r-1}\}\) for \(m\) its index and \(r\) its period
Page: 11
Section: 2
Variables: \(S\) a semigroup,
\(a\in S\)
Notation: \(K_a\)
Typ: A cyclic subgroup and an ideal of \(\langle a\rangle\)
Assuming: \(S\) is monogenic

Smaller0: there exists an index \(a\), a period \(r\):
-\(a^m=a^{m+r}\)
-\(a^{m+u}=a^{m+v}\) iff \(u\equiv v\mod r\),
-\(\langle a\rangle=\{a^i\mid 1\le i<m+r\}\)
-\(K_a\) is cyclic subgroup of \(\langle a\rangle\)
Greater0: \(\langle a\rangle\) is isomorphic to \((\mathbb N,+)\)
Page: 12
Chapter: 1
Section: 2
Variables: \(S\) a semigroup,
\(a\in S\)
Kind: Theo
Smaller to greater: xor

Name: A periodic semigroup
Definition: A semigroup whose elements's order is finite
Page: 12
Section: 2
Index: 3
Kind: Prop
Extra: example

Smaller0: \(S\) is finite
Greater0: \(S\) is periodic
Page: 12
Chapter: 1
Section: 2
Variables: \(S\) a semigroup
Index: 3
Kind: Prop

Smaller0: \(S\) is periodic
Greater0: an idempotent power
Page: 12
Chapter: 1
Section: 2
Variables: \(S\) a semigroup,
\(a\in S\)
Index: 3
Kind: Prop
Prefix Greater: \(a\) has
Extra: Scis finit e

Question: The number of minimum is
Answer: at most one
Page: 13
Chapter: 1
Section: 3
Variables: \((X,\le)\) a partially ordered set
Kind: Prop

Name: A minimal element of \(X\)
Definition: An element \(a\in X\) such that there is no \(b\in X\) such that \(b\le a\)
Page: 13
Section: 3
Variables: \((X,\le)\) a partially ordered set

Name: A transitive relation on \(X\)
Definition: A binary relation which contains \((x,z)\) when it contains \((x,y)\) and \((y,z)\)
Page: 13, 19
Section: 3
Variables: \((X,\le)\) a partially ordered set
Definition2: A relation \(\rho\) such that \(\rho\circ\rho\subseteq\rho\)

Name: A partial order
Definition: reflexive
Page: 13
Section: 3
Definition2: transitive
Conjdef: A binary relation
Definition3: antisymmetric

Name: A reflexive relation on \(X\)
Definition: which contains \((x,x)\) for each \(x\in X\)
Page: 13, 19
Section: 3
Variables: \(X\) a set
Definition2: wich is a superset of \(1_X\)
Conjdef: A binary relation

Name: A symmetric relation on \(X\)
Definition: which contains \((x,y)\) iff it contains \((y,x)\)
Page: 19
Section: 3
Variables: \(X\) a set
Definition2: such that \(\rho=\rho^{-1}\)
Conjdef: A binary relation \(\rho\) such that, equivalently

Name: A total order on \(X\)
Definition: A partial order such that \(x\le y\) or \(y\le x\) for each \(x,y\in X\)
Page: 13
Section: 3
Variables: \(X\) a set

Name: A totally(partially) ordered set
Definition: A pair \((X,\le)\) with \(X\) a set and \(\le\) a total(partial) order on \(X\)
Page: 13
Section: 3

Name: A minimum of \(X\)
Definition: An element \(a\in X\) such that for all \(b\in X\), \(a\le b\)
Page: 13
Section: 3
Variables: \((X,\le)\) a partially ordered set

Name: A lower bound of \(Y\)
Definition: An element \(c\in X\) such that \(c\le y\) for every \(y\in Y\)
Page: 13
Section: 3
Variables: \((X,\le)\) a partially ordered set,
\(Y\subseteq X\)

Name: The meet of \(Y\)
Definition: The maximal element of the lower bound of \(Y\)
Page: 14
Section: 3
Variables: \((X,\le)\) a partially ordered set,
\(Y\subseteq X\)
Notation: \(\bigwedge Y\)
Name2: the greatest lower bound of \(Y\)
Assuming: The set of lower bound of \(Y\) contains a maximal element

Page: 14
Section: 3
Notation: \(a\land b\)
Variables: \((X,\le)\) a partially ordered set,
\(a,b\in X\)
Chapter: 1
Definition: \(\bigwedge\{a,b\}\)
Kind: Prop
Index: 2
Commutative: commutative
Associative: Associative
Idempotent: idempotent

Name: A lower semilattice
Definition: An ordered set \((X,\le)\) such that \(a\land b\) exists for each \(a,b\in X\)
Page: 14
Section: 3
Denoted by: \((X,\le,\land)\)

Name: A complete lower semilattice
Definition: An ordered set \((X,\le)\) such that \(\bigwedge Y\) exists for each \(Y\subseteq X\)
Page: 14
Section: 3

Smaller0: \(a\land b=a\)
Greater0: \(a\le b\)
Page: 14
Chapter: 1
Section: 3
Variables: \((X,\le)\) a complete lower semilattice,
\(a,b\in X\)
Smaller to greater: iff

Name: A maximum of \(X\)
Definition: An element \(a\in X\) such that for all \(b\in X\), \(b\le a\)
Page: 13
Section: 3
Variables: \((X,\le)\) a partially ordered set

Name: A greater bound of \(Y\)
Definition: An element \(c\in X\) such that \(y\le c\) for every \(y\in Y\)
Page: 13
Section: 3
Variables: \((X,\le)\) a partially ordered set,
\(Y\subseteq X\)

Name: The join of \(Y\)
Definition: The minimal element of the upper bound of \(Y\)
Page: 14
Section: 3
Variables: \((X,\le)\) a partially ordered set,
\(Y\subseteq X\)
Notation: \(\bigvee\{y\mid y\in Y\}\)
Name2: the least upper bound of \(Y\)
Assuming: The set of upper bound of \(Y\) contains a maximal element

Page: 14
Section: 3
Notation: \(a\lor b\)
Variables: \((X,\le)\) a partially ordered set,
\(a,b\in X\)
Chapter: 1
Definition: \(\bigvee\{a,b\}\)
Kind: Prop
Index: 2
Commutative: commutative
Associative: Associative
Idempotent: idempotent

Name: An upper semilattice
Definition: An ordered set \((X,\le)\) such that \(a\lor b\) exists for each \(a,b\in X\)
Page: 14
Section: 3
Denoted by: \((X,\le,\lor)\)

Name: A complete upper semilattice
Definition: An ordered set \((X,\le)\) such that \(\bigvee \{y\mid y\in Y\}\) exists for each \(Y\subseteq X\)
Page: 14
Section: 3

Smaller0: \(a\lor b=a\)
Greater0: \(b\le a\)
Page: 14
Chapter: 1
Section: 3
Variables: \((X,\le)\) a complete lower semilattice,
\(a,b\in X\)
Smaller to greater: iff

Name: A (complete) semilattice
Definition: A (complete) lower and upper semilattice
Page: 14
Section: 3
Denoted by: \((X,\le,\land,\lor)\)

Name: A sublattice of \(X\)
Definition: A nonempty subset of \(X\), closed under \(\land\) and \(\lor\)
Page: 14
Section: 3
Variables: \(X=(X,\le,\land,\lor)\) a lattice

Page: 15
Chapter: 1
Section: 3
Name: The semigroup associated to \(X\)
Set: \(X\)
Variables: \(X=(X,\le)\) a lower-semilattice
*: \(\land\)

Name: Its Hasse diagram
Definition: A diagram, with each element of \(X\) represented by a dot, a line from \(a\) to \(b\), with \(a\) below \(b\), if \(a\le b\) and there is no \(a\le c\le b\).
Page: 16
Section: 4
Variables: \((X,\le)\) a partially ordered set,

Name: A binary relation on \(X\)
Definition: A subset of \(X\times X\)
Page: 16
Section: 4
Variables: \(X\) a set
Denoted by: \(\rho\)

Page: 16
Definition: \(X\times X\)
Chapter: 1
Section: 4
Variables: \(X\) a set
Name: The universal relation
Context: A relation

Page: 16
Definition: \(\{(x,x)\mid x\in X\}\)
Chapter: 1
Section: 4
Variables: \(X\) a set
Notation: \(1_X\)
Name: The equality relation
Context: a relation

Page: 16
Section: 4
Notation: \(\circ\)
Variables: \(X\) a set
Chapter: 1
Definition: \((\rho,\sigma)\mapsto \{(x,z)\in X^2\mid \exists z\in x; (x,z)\in \rho, (z,y)\in \sigma\}\)
Associative: associative
Context: on binary relations

Name: The set of all binary relations on \(X\)
Page: 16
Section: 4
Variables: \(X\) a set
Notation: \(\mathcal B_X\)
Kind: Prop

Page: 16
Definition: The \(=\) relation if \(n=0\),
\(\rho^{n-1}\circ\rho\) if \(n>0\)
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\rho\in\mathcal B_X\),
\(n\in\mathbb N\)
Notation: \(\rho^{n}\)

Name: The domain of \(\rho\)
Definition: the set of \(x\in X\) such that there exists \(y\in X\) such that \((x,y)\in\rho\)
Page: 17
Section: 4
Variables: \(X\) a set,
\(\rho\in\mathcal B_X\),
Abbreviation: \(\text{dom} \rho\)

Definition: the set of \(x\in X\) such that there exists \(y\in X\) such that \((y,x)\in\rho\)
Page: 17
Section: 4
Chapter: 1
Variables: \(X\) a set,
\(\rho\in\mathcal B_X\),
Name: The image of \(\rho\)
Monotonic: Monotone function
Abbreviation: \(\text{im }\rho\)

Definition: the function sending \(\rho\) to \(\{(x,y)\mid(y,x)\in \rho\}\)
Page: 17
Section: 4
Notation: \(\cdot^{-1}\)
Index: ,(1.4.9)
Kind: inverse monotonic, involution

Question: \((\rho_1\circ\dots\circ \rho_n)^-1\)
Answer: equals \((\rho_n^{-1}\circ\dots\circ\rho_1^{-1})\)
Page: 17
Chapter: 1
Section: 4
Index: ,(1.4.9)

Question: \(\text{dom}(\rho^{-1})\)
Answer: equals \(\text{im}\rho\)
Page: 17
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\rho\) a relation

Question: \(\text{im}(\rho^{-1})\)
Answer: equals \(\text{dom}\rho\)
Page: 17
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\rho\) a relation

Name: A partial map
Definition: An element \(\phi\in\mathcal B_X\) such that \(|x\phi|=1\) for all \(x\in\text{dom}\phi\)
Page: 17
Section: 4
Variables: \(X\) a set,

Name: \(\phi\) is a restriction of \(\psi\)
Definition: \(\phi\subseteq\psi\)
Page: 17
Section: 4
Variables: \(X\) a set,
\(\phi,\psi\in\mathcal B_X\) partial maps
Name2: \(\psi\) is an extension of \(\phi\)

Name: \(\phi\) restricted to \(A\)
Definition: \(\{(x,x\phi)\mid x\in A\}\)
Page: 17
Section: 4
Variables: \(X\) a set,
\(\phi\in\mathcal B_X\) a partial map
\(A\subseteq \text{dom}\phi\)
Notation: \(\phi_{|A}\)

Definition: The set of partial maps
Page: 18
Section: 4
Variables: \(X\) a set,
Notation: \(\mathcal P_X\)
Typ: A subsemigroup of \(\mathcal B_X\)

Name: \(\mathcal P_X\)
Page: 18
Section: 4
Chapter: 1
Index: 2
Variables: \(X\) a set,
Kind: Prop
Under: product

Question: \(\text{dom}(\phi\circ\psi)\)
Answer: equals \([\text{im}\ \phi\cap\text{dom}\ \psi]\phi^-1\)
Page: 18
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\phi,\psi\in\mathcal P_X\)
Index: 3
Kind: Prop

Question: \(\text{im}(\phi\circ\psi)\)
Answer: equals \([\text{im}\ \phi\cap\text{dom}\ \psi]\psi\)
Page: 18
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\phi,\psi\in\mathcal P_X\)
Index: 3
Kind: Prop

Smaller0: \(x(\phi\circ\psi)\)
Greater0: \((x\phi)\psi\)
Page: 18
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\phi,\psi\in\mathcal P_X\)
Index: 3
Kind: Prop
Smaller to greater: =

Name: A map
Definition: A partial map whose domain is \(X\)
Page: 19
Section: 4
Variables: \(X\) a set,
Name2: A function

Name: The set of function on \(X\)
Page: 19
Section: 4
Variables: \(X\) a set,
Notation: \(\mathcal T_X\)
Kind: Prop
Typ: Subsemigroup of \(\mathcal B_X\) (the set of relations on \(X\))

Name: An antisymmetric relation on \(X\)
Definition: which never contains both \((x,y)\) and \((y,x)\) for \(x\ne y\)
Page: 13,19
Section: 3
Variables: \(X\) a set
Definition2: such that \(\rho\cap\rho^{-1}\subseteq 1_X\)
Conjdef: A binary relation \(\rho\)

Name: A rectangular band
Definition: \(aba=a\) for each elements \(a,b\)
Page: 7
Index: 3
Kind: Theo
Definition2: \(S\) is a band and \(abc=ac\) for each \(a,b,c\in S\)
Conjdef: A semigroup \(S\) such that, equivalently
Definition3: \(S\) is the product of a left zero semigroup and of a right zero semigroup,
Definition4: \(S\) is of the form \((A\times B, ((a,b),(a',b'))\mapsto (a,b'))\).

Smaller0: arbitrary
Greater0: \(S\)
Page: 9
Chapter: 1
Section: 2
Smaller to greater: contains
Greater to smaller: belongs to
Prefix Smaller: The set of semigroup/submonoid of \(S\) containing \(A\) is:
-closed under
Suffix Smaller: intersection

Smaller0: \(\rho_1\subseteq\rho_2\)
Greater0: \(\text{dom}\rho_1\subseteq\text{dom}\rho_2\)
Page: 17
Chapter: 1
Section: 4
Variables: \(X\) a set,
\(\rho_1,\rho_2\in\mathcal B_X\),

Name: A set satisfying the minimal condition
Definition: An ordered set whose every nonempty subset admits a minimal element
Page: 13
Section: 3

Name: A well ordered set
Definition: satisfying the minimal condition
Page: 13
Section: 3
Definition2: totally ordered
Conjdef: An ordered set

Question: The map [$]\defFun{S}{\mathcal T_{S^1}}{a}{\rho_a}[/$] is
Answer: faithful
Page: 7
Chapter: 1
Index: 2
Kind: Theo

Name: Right multiplication by \(a\)
Definition: [$]\defFun{S^1}{S^1}{x}{x a}[/$]
Page: 7
Variables: \(S\) a semigroup
\(a\in S^1\)
Notation: \(\rho_a\)
Index: 2
Kind: Theo

Name: The semilattice obtained from \(X\)
Definition: The lower semilattice with \(a\le b\) defined as \(a\cdot b=a\)
Page: 15
Section: 3
Variables: \((X,\cdot)\) a semigroup
Index: 2
Kind: Prop
Intuition: . represents min
Assuming: each elements of \(X\) are idempotent

Smaller0: belongs to \(\mathcal P_X\)
Greater0: one-one
Page: 19
Chapter: 1
Section: 4
Variables: \(X\) a set
\(\phi\in\mathcal P_X\)
Smaller to greater: iff
Prefix Greater: \(\phi\) is
Prefix Smaller: \(\phi^{-1}\)

Smaller0: belongs to \(\mathcal T_X\)
Greater0: bijective
Page: 19
Chapter: 1
Section: 4
Variables: \(X\) a set
\(\phi\in\mathcal P_X\)
Smaller to greater: iff
Prefix Greater: \(\phi\) is
Prefix Smaller: \(\phi^{-1}\)

Name: A \(\rho\)-class
Definition: A set \(x\rho\) for some \(x\in X\)
Page: 20
Section: 4
Variables: \(X\) a set
\(\rho\) an equivalence relation on \(X\)
Notation: \(X/\rho\)
Index: 6
Kind: Prop
Name2: A \(\rho\) equivalence-class

Name: The quotient set of \(X\) by \(\rho\)
Definition: the set of \(\rho\)-classes
Page: 20
Section: 4
Variables: \(X\) a set
\(\rho\) an equivalence relation on \(X\)
Notation: \(X/\rho\)
Index: 6
Kind: Prop
Typ: A partition of \(X\)

Name: The equivalence relation associated with \(\pi\)
Definition: the relation containing \((x,y)\) iff \(x,y\) both belong to some \(A_i\)
Page: 20
Section: 4
Variables: \(X\) a set
\(\pi=\{A_i\}\) a partition of \(X\)
Index: 6
Kind: Prop

Definition: \((x,y)\in\rho\)
Page: 20
Section: 4
Variables: \(X\) a set
\(\rho\) an equivalence relation on \(X\)
Notation: \(x\rho y\)
Index: 6
Kind: Prop
Notation2: \(x\equiv y(\mod p)\)

Name: The natural map for \(\rho\)
Definition: [$]\defFun{X}{X/\rho}{x}{x\rho}[/$]
Page: 20
Section: 4
Variables: \(X\) a set
\(\rho\) an equivalence relation on \(X\)
Notation: \(\rho^\natural\)
Index: 6
Kind: Prop

Smaller0: a map
Greater0: an equivalence relation
Page: 20
Chapter: 1
Section: 4
Variables: \(X,Y\) sets
\(\rho\subseteq X\times Y\) a map
Index: 6
Kind: Prop
Prefix Greater: \(\phi\circ\phi^{-1}\) is
Prefix Smaller: \(\rho\) is