Title: Theory / Unimportant / Math / Al gebra / Universal / Burris-Sankappanavar+16{ / Intro
Tags: récupéré hasSeenConjDef examplesSwitchExample1 toBinaryFunction toBinaryRelation

Page: 5, 8
Section: 1
Notation: \(\lor\)
Chapter: I
Name: join
Example: or
Commutative: commutative
Associative: associative
Idempotent: idempotent
Applied to: In propositional logic
Example2: lcm
Applied to2: natural number and division
Applied to3: lattice
Example3: least upper bound

Page: 5, 8
Section: 1
Notation: \(\land\)
Chapter: I
Name: meet
Example: and
Commutative: commutative
Associative: associative
Idempotent: idempotent
Applied to: logic
Example2: greatest common divisor
Applied to2: natural number and division
Applied to3: lattice
Applied to4: sets
Example4: intersection

Name: Absorption law
Definition: \(x\equiv x\lor(x\land y)\) and \(x\equiv x\land(x\lor y)\)
Page: 6
Chapter: I
Section: 1

Name: Lattice
Definition: A tuple \((L,\land,\lor)\) with \(\land\), \(\lor\) associative, commutative, idempotent satisfying the absorption law
Page: 5, 8, 11
Chapter: I
Section: 1
Index: 1, 4, 1
Kind: Def, Def, Ex
Definition2: A poset with sup and inf defined
Example: Closed (or open) set in a topological space, with inclusion

Name: Partially ordered set
Page: 5
Chapter: I
Section: 1
Index: 2
Kind: Def
Abbreviation: poset

Name: A chain
Page: 5
Chapter: I
Section: 1
Kind: Def
Name2: A totally ordered set
Name3: linearly ordered set

Name: The power set of \(A\)
Definition: elements are subset of \(A\)
Page: 5
Chapter: I
Section: 1
Variables: \(A\) a set
Notation: \(\mathrm{SU}(A)\)
Index: (1)
Kind: Ex
Definition2: order: inclusion
Conjdef: Partially ordered sets

Name: An upper bound of \(A\)
Definition: An element \(a\in P\) such that \(p\le a\) for each \(p\in P\)
Page: 5
Chapter: I
Section: 1
Variables: \(P\) a poset
\(A\) a subset of \(P\)
Context: hesse diagram
Index: 3
Kind: Def
Representation: An element above each element of \(P\) in the Hesse diagram

Name: A lower bound of \(A\)
Definition: An element \(a\in P\) such that \(p\ge a\) for each \(p\in P\)
Page: 6
Chapter: I
Section: 1
Variables: \(P\) a poset
\(A\) a subset of \(P\)
Index: 3
Kind: Def
Representation: An element below each element of \(P\) in the Hesse diagram

Name: A greatest lower bound of \(A\)
Page: 6
Chapter: I
Section: 1
Variables: \(P\) a poset
\(A\) a subset of \(P\)
Index: 3
Kind: Def
Name2: An infimum of \(A\)
Abbreviation: g.l.b. of \(A\)
Abbreviation2: inf \(A\)

Name: A least upper bound of \(A\)
Definition: A minimal upper bound of \(A\)
Page: 6
Chapter: I
Section: 1
Variables: \(P\) a poset
\(A\) a subset of \(P\)
Index: 3
Kind: Def
Name2: A supremum of \(A\)
Abbreviation: l.u.b. of \(A\)
Abbreviation2: sup \(A\)

Name: A cover of \(a\)
Definition: a least element strictly greater than \(a\)
Page: 7
Chapter: I
Section: 1
Variables: \(P\) a poset
\(a\in A\)
Context: poset
Index: 3
Kind: Def

Name: A is covered by \(a\)
Definition: a greatest element strictly smaller than \(a\)
Page: 7
Chapter: I
Section: 1
Variables: \(P\) a poset
\(a\in A\)
Index: 3
Kind: Def

Page: 7
Chapter: I
Section: 1
Variables: \(P\) a poset
\(a,b\in A\)
Notation: \(b\prec a\)
Name: \(a\) covers \(b\)
Kind: Def
Index: 3

Name: The Hasse diagram of \(P\)
Definition: nodes: elements of \(P\)
Page: 7
Chapter: I
Section: 1
Variables: \(P\) a poset
Index: 3
Kind: Def
Definition2: arrow from \(a\) to \(b\) if \(b\) covers \(a\)
Conjdef: A digraph with
Representation: \(a\) covers \(b\) is represented by \(a\) above \(b\)

Definition: \(a=a\land b\), equivalently \(b=a\lor b\)
Page: 8
Chapter: I
Section: 1
Variables: \((L,\land,\lor)\) a lattice
\(a,b\in L\)
Notation: \(a\le b\)
Index: 4
Kind: Def

Name: A morphism from \(L_1\) to \(L_2\)
Definition: A function \(\alpha:L_1\to L_2\) such that \(\alpha(a\land b)=\alpha(a)\land\alpha(b)\) and \(\alpha(a\lor b)=\alpha(a)\lor\alpha(b)\)
Page: 10
Chapter: I
Section: 2
Variables: \((L_i,\land_i,\lor_i)\) lattices
Index: 4
Kind: Def

Name: An isomorphism from \(L_1\) to \(L_2\)
Definition: A bijective morphism from \(L_1\) to \(L_2\)
Page: 10
Chapter: I
Section: 2
Variables: \((L_i,\land_i,\lor_i)\) lattices
Index: 4
Kind: Def

Page: 10
Definition: Lattices such that, equivalently there is an isomorphism from one to the other
Chapter: I
Section: 2
Name: Isomorphic lattices
Kind: Def, Theo
Index: 1, 3
Definition2: Lattices such that, equivalently there is \(\alpha:L_1\to L_2\) order-preserving whose inverse is order preserving

Name: Order preserving between \(P_i\)
Definition: A function \(\alpha:P_1\to P_2\) such that \(a\le b\) implies \(\alpha(a)\le\alpha(b)\)
Page: 10
Chapter: I
Section: 2
Variables: \((P_i,\le_i)\) posets
Kind: Def
Extra: monotonic

Smaller0: \(\alpha\) has an inverse order preserving
Greater0: \(\alpha\) is bijective
Page: 11
Chapter: I
Section: 2
Variables: \((P_i,\le_i)\) posets
\(\alpha:P_1\to P_2\) order preserving
Kind: Def
Context: Poset mapping

Name: A sublattice of \(L\)
Definition: A subset of \(L\) closed under \(\land\) and \(\lor\)
Page: 11
Chapter: I
Section: 2
Variables: \((L,\lor,\land)\) a lattice
Index: 4
Kind: Def

Name: \(L_1\) can be embedded into \(L_2\)
Definition: \(L_1\) is isomorphic to a sublattice of \(L_2\)
Page: 11
Chapter: I
Section: 2
Variables: \((L_1,\lor_1,\land_1)\) a lattice
Index: 5
Kind: Def

Name: A distributive lattice
Definition: \(x\land(y\lor z)\equiv (x\land y)\lor (x\land z)\) or \(x\lor(y\land z)\equiv (x\lor y)\land(x\lor z)\)
Page: 12
Chapter: I
Section: 3
Index: 1
Kind: Ex
Definition2: neither \((M_5)\) nor \((N_5)\) can be embedded into \(L\)
Conjdef: A lattice such that, equivalently

Smaller0: \(x\land(y\lor z)\equiv (x\land y)\lor (x\land z)\)
Greater0: \(x\lor(y\land z)\equiv (x\lor y)\land(x\lor z)\)
Page: 12
Chapter: I
Section: 3
Variables: \(L\) a lattice
Index: 1
Kind: Ex
Smaller to greater: iff
Hide sides: yes

Name: A modular lattice
Definition: A lattice satisfying the modular law
Page: 13
Chapter: I
Section: 3
Kind: Def
Definition2: \(N_5\) can not be embedded in it

Name: The modular law
Definition: \(w\le y\) implies \(w\lor (y\land z)\equiv y\land (w\lor z)\)
Page: 13
Chapter: I
Section: 3
Variables: \(L\) a lattice
Kind: Def
Definition2: \((x\land y)\lor (y\land z)\equiv y\land((x\land y)\lor z)\)
Construction: w=x land y, hence w le y

Smaller0: \(L\) is distributive
Greater0: \(L\) is modular
Page: 13
Chapter: I
Section: 3
Variables: \(L\) a lattice
Kind: Def
Counterexample: M5

Page: 13
Chapter: I
Section: 3
Notation: \(M_5\)
Kind: Def
Typ: Modular, Not distributive

Page: 13
Chapter: I
Section: 3
Notation: \(N_5\)
Kind: Def
Property: Not modular, Not distributive

Smaller0: \(L\) is modular
Greater0: \((N_5)\) can be embedded into \(L\)
Page: 14
Chapter: I
Section: 3
Index: 5
Kind: Theo
Smaller to greater: xor
Name: Dedekind law

Smaller0: \(L\) is nondistributive
Greater0: \((N_5)\) or \((M_5)\) can be embedded into \(L\)
Page: 14
Chapter: I
Section: 3
Index: 5
Kind: Theo
Smaller to greater: iff
Name: Birkhoff law

Question: \(A^0\)
Answer: equals \(\{\emptyset\}\)
Page: 25
Chapter: II
Section: 1
Variables: \(A\) a set

Definition: The set of \(n\)-tuples of elements from \(A\)
Page: 25
Chapter: II
Section: 1
Variables: \(A\) a set,
\(n\in\mathbb N\)
Notation: \(A^n\)

Name: An \(n\)-ary operation
Definition: A function \(A^n\to A\)
Page: 25
Chapter: II
Section: 1
Variables: \(A\) a set,
\(n\in\mathbb N\)

Name: A finitary operation
Definition: An \(n\)-ary operation for some \(n\in\mathbb N\)
Page: 25
Chapter: II
Section: 1
Variables: \(A\) a set,
\(n\in\mathbb N\)

Name: A nullary operation
Definition: A function of arity \(0\)
Page: 25
Chapter: II
Section: 1
Name2: A constant

Name: A unary operation
Definition: An operaton of arity \(1\)
Page: 26
Chapter: II
Section: 1
Variables: \(A\) a set

Name: A binary operation
Definition: An operaton of arity \(2\)
Page: 26
Chapter: II
Section: 1
Variables: \(A\) a set

Name: A ternary operation
Definition: An operaton of arity \(3\)
Page: 26
Chapter: II
Section: 1
Variables: \(A\) a set
Kind: Def

Name: A language of algebra
Definition: of function symbols
Page: 26
Chapter: II
Section: 1
Index: 2
Kind: Def
Definition2: such that each symbol as a non-negative number called its arity
Conjdef: A set
Name2: A type of algebra
Denoted by: \(\mathcal F\)

Definition: The set of \(n\)-ary function symbols of \(\mathcal F\)
Page: 26
Chapter: II
Section: 1
Variables: \(\mathcal F\) a language
Notation: \(\mathcal F_n\)
Index: 2
Kind: Def

Name: The arity of \(f\)
Definition: A number \(n\) such that \(f\) is \(n\)-ary
Page: 26
Chapter: II
Section: 1
Variables: \(\mathcal F\) a language
\(f\in\mathcal F\)
Index: 2
Kind: Def
Name2: The rank of \(f\)

Name: An algebra of type \(\mathcal F\)
Definition: \(A\) a nonempty set,
Page: 26
Chapter: II
Section: 1
Variables: \(\mathcal F\) a language of algebra
Index: 3
Kind: Def
Definition2: \(F\) a family of finitary operations which contains
Conjdef: A tuple
Definition3: \(f^{\mathrm A}:A^n\to A\) for each \(f\in\mathcal F_n\)
Denoted by: \(\mathbf A\)

Name: A fundamental operation of \(\mathbf A\)
Definition: Some \(f^{\mathbf A}\) for \(f\in\mathcal F\)
Page: 26
Chapter: II
Section: 1
Variables: \(\mathcal F\) a language of algebra
\(\mathbf A=\langle A,F\rangle\) an algebra of type \(\mathcal F\)
Index: 3
Kind: Def

Definition: A finite language of algebra
Page: 26
Chapter: II
Section: 1
Index: 3
Kind: Def
Denoted by: \(\mathcal F=\{f_1,\dots,f_k\}\)
Assuming: the arity are in non-decreasing order

Definition: \(\langle A,F\rangle\), with the implicity identity between \(f_i\) and \(f_i^{\mathrm A}\)
Page: 26
Chapter: II
Section: 1
Variables: \(\mathcal F=\{f_1,\dots,f_k\}\)
Notation: \(\langle A,f_1,\dots,f_k\rangle\)
Index: 3
Kind: Def

Name: A unary algebra
Definition: An algebra whose all of its operations are unary
Page: 26
Chapter: II
Section: 1
Index: 3
Kind: Def

Name: A mono-unary algebra
Definition: An algebra with a single unary operation
Page: 26
Chapter: II
Section: 1
Index: 3
Kind: Def

Name: A groupoid
Definition: An algebra with a single binary operation
Page: 26
Chapter: II
Section: 1
Kind: Def

Name: A trivial algebra
Definition: An algebra whose set's cardinal is 1
Page: 26
Chapter: II
Section: 1
Kind: Def

Name: A finite algebra
Definition: An algebra whose set's cardinal is finite
Page: 26
Chapter: II
Section: 1
Kind: Def

Name: A morphism from \(\mathbf A\) to \(\mathbf B\)
Definition: A function \(\alpha:A\to B\) such that \(\alpha f^{\mathbf A}(a_1,\dots, a_n)=f^{\mathbf{B}}(\alpha a_1,\dots,\alpha a_n)\) for each \(f\in\mathcal F\) and \(a_1,\dots,a_n\in A\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}\) two algebras of type \(\mathcal F\)
Index: 1
Kind: Def

Name: An isomorphism from \(\mathbf A\) to \(\mathbf B\)
Definition: one-to-one and onto
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}\) two algebras of type \(\mathcal F\)
Index: 1
Kind: Def
Definition2: which is a morphism
Conjdef: A function \(\alpha:A\to B\)

Name: Isomorphic algebras
Definition: Alegbras such that there exists an isomorphism between them
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
Index: 1
Kind: Def

Name: A subuniverse of \(\mathbf A\)
Definition: A subseteq \(B\subseteq A\) closed under each \(f\in\mathcal F\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A}=\langle A,\mathcal F\rangle\) an algebras of type \(\mathcal F\)
Kind: Def

Name: A subalgebra of \(\mathbf A\)
Definition: whose set is a subuniverse of \(\mathbf A\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A}=\langle A,\mathcal F\rangle\) an algebras of type \(\mathcal F\)
Kind: Def
Definition2: with \(f^{\mathbf B}=f^{\mathbf A}_{\mid B}\)
Conjdef: An algebra \(\mathbf B\) of type \(\mathcal F\)

Name: \(\mathbf B\) is a subalgebra of \(\mathbf A\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}\) algebras of type \(\mathcal F\)
Notation: \(\mathbf{B\le A}\)
Kind: Def

Smaller0: \(\mathbf B\) is a subalgebra of \(\mathbf A\)
Greater0: \(B\) is a subuniverse of \(\mathbf A\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}\) algebras of type \(\mathcal F\)
Kind: Def

Smaller0: the set of subalgebra of \(G\)
Greater0: the set of subsemigroup of \(G\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F=\{\cdot\}\) a type
\(\mathbf G=\langle G,\cdot^{\mathbf G}\rangle\) a group
Kind: Def
Smaller to greater: equals

Smaller0: the set of subalgebra of \(G\)
Greater0: the set of subgroup of \(G\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F=\{\cdot,^{-1}\}\) a type
\(\mathbf G=\langle G,\cdot^{\mathbf G}\rangle\) a group
Kind: Def
Smaller to greater: equals

Question: Why is it important to add each constant and operation in the vocabulary, instead of considering it using axioms ?
Answer: Because subalegbra only ensures closure under functions of the vocabulary. A subalgebra of a group is not a group if the vocabulary does not contain \(^{-1}\)
Page: 31
Chapter: II
Section: 2
Context: universal algebra

Name: An embedding of \(\mathbf{A}\) into \(\mathbf B\)
Definition: one-to-one
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A}=\langle A,\mathcal F\rangle\) an algebras of type \(\mathcal F\)
Index: 3
Kind: Def
Definition2: morphism
Conjdef: A function
Name2: A monomorphism

Name: An algebra which can be embedded into \(\mathbf A\)
Definition: An algebra \(\mathbf B\) with an embedding from \(\mathbf B\) into \(\mathbf A\)
Page: 31
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A}=\langle A,\mathcal F\rangle\) an algebras of type \(\mathcal F\)
Index: 3
Kind: Def

Smaller0: \(\alpha\) is an embedding
Greater0: is a subuniverse of \(\mathbf B\)
Page: 32
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}=\langle A,\mathcal F\rangle\) algebras of type \(\mathcal F\),
\(\alpha:\mathbf{A\to B}\)
Index: 4
Kind: Theo
Prefix Greater: \(\alpha(A)\)

Definition: The subalgebra of \(\mathbf B\) with universe \(\alpha(A)\)
Page: 32
Chapter: II
Section: 2
Variables: \(\mathcal F\) a type
\(\mathbf{A,B}=\langle A,\mathcal F\rangle\) algebras of type \(\mathcal F\),
\(\alpha:\mathbf{A\to B}\) an embedding
Notation: \(\alpha(\mathbf A)\)
Index: 5
Kind: Def

Name: Main question of algebraist
Definition: Is every member of \(K\) isomorphic to some member of a subclass \(K_1\)
Page: 32
Chapter: II
Section: 2
Variables: \(K\) a class of algebra,
Index: 5
Kind: Def
Definition2: Is every member of \(K\) embedable into some member of a subclass \(K_1\)

Name: Complementation
Page: 32
Chapter: IV
Section: 1
Context: Boolean algebras
Notation: \('\)
Kind: Def