Title: Theory / Unimportant / Math / Al gebra / Universal / Burris-Sankappanavar+16{ / Examples=
Tags: récupéré MissingMedia

Page: 27
Section: 1
Chapter: II
Axiom: \(x\backslash(x\cdot y)\equiv y\) and \((x\cdot y)/y\equiv x\)
Name: A quasigroup
Vocabulary: \(/,\cdot,\backslash\)
Kind: Def
Axiom2: \(x\cdot (x\backslash y)\equiv y\) and \((x/y)\cdot y\equiv x\)

Page: 27
Section: 1
Chapter: II
Axiom: \(1\) is an identity for \(\cdot\)
Name: A loop
Vocabulary: \(\langle/,\cdot,\backslash,1\rangle\)
Extends: Theory of quasigroups

Page: 27
Section: 1
Chapter: II
Axiom: \(f_r(x+y)\equiv f_r(x)+f_r(y)\) for \(r\in R\)
Name: Theory of left \(R\)-modules
Vocabulary: \(\langle +,-,0,(f_r)_{r\in R}\rangle\) with \(f_r\) unary
Extends: Abelian group for \(\langle +,-,0\rangle\)
Kind: Ex
Index: 5
Axiom2: \(f_{r+s}(x)\equiv f_r(x)+f_s(x)\) for \(r,s\in R\)
Axiom3: \(f_r(f_s(x))\equiv f_{rs}(x)\) for \(r,s\in R\)
Variables: \(R\) a ring

Page: 27
Section: 1
Chapter: II
Axiom: \(f_1(x)\equiv x\)
Name: Unitary left-\(R\)-module
Vocabulary: \(\langle +,-,0,(f_r)_{r\in R}\rangle\) with \(f_r\) unary
Extends: Left \(R\)-module
Variables: \(R\) a ring with unity

Page: 28
Section: 1
Chapter: II
Axiom: \(f_r(x\cdot y)\equiv (f_r(x))\cdot y\equiv x\cdot f_r(y)\) for \(r\in R\)
Name: Algebra over \(R\)
Vocabulary: \(\langle +,-,\cdot, 0,(f_r)_{r\in R}\rangle\) with \(f_r\) unary
Extends: Left \(R\)-module for \(\langle +,-,0,(f_r)\rangle\)
Kind: Ex
Index: 6
Variables: \(R\) a ring with identity
Extends2: Ring for \(\langle +,\cdot,-,0\rangle\)

Page: 28
Section: 1
Chapter: II
Axiom: \(x\cdot x\equiv x\)
Name: Idempotent semigroup
Vocabulary: \(\langle\cdot\rangle\)
Extends: Semigroup
Kind: Ex
Index: 7
Extra: boolean ?

Page: 28
Section: 1
Chapter: II
Name: Semilattice
Vocabulary: \(\langle\cdot\rangle\)
Extends: Commutative semigroup
Kind: Ex
Index: 7
Extends2: Idempotent semigroup

Page: 28
Section: 1
Chapter: II
Axiom: Absorption law
Name: Lattice
Vocabulary: \(\langle\lor,\land\rangle\)
Extends: Semilattice on \(\land\) and on \(\lor\)
Kind: Ex
Index: 8

Page: 28
Section: 1
Chapter: II
Axiom: \(x\land 0\equiv 0\), \(x\lor 1\equiv 1\)
Name: Bounded lattice
Vocabulary: \(\langle\lor,\land,0,1\rangle\)
Extends: Lattice
Kind: Ex
Index: 9

Page: 28
Section: 1
Chapter: II
Axiom: Each binary operation is distributive over the other one
Name: Distributive lattice
Vocabulary: \(\langle\lor,\land,0,1\rangle\)
Extends: Lattice
Kind: Ex
Index: 10

Page: 28
Section: 1
Chapter: II
Axiom: \(x\land x\equiv 0\) and \(x\lor x'\equiv 1\)
Name: Boolean algebra
Vocabulary: \(\langle\lor,\land,',0,1\rangle\)
Extends: Distributive lattice
Kind: Ex
Index: 10
Axiom8: BA
Axiom9: Bounded lattice

Page: 28
Section: 1
Chapter: II
Axiom: \(x\to x\equiv 1\)
Name: Heyting algebra
Vocabulary: \(\langle\lor,\land,\to,0,1\rangle\)
Extends: Distributive bounded lattice
Kind: Ex
Index: 11
Axiom2: \(x\to(y\land z)\equiv (x\to y)\land (x\to z)\) and \((x\lor y)\to z\equiv (x\to z)\lor (y\to z)\)
Axiom3: \((x\to y)\land y\equiv y\) and \(x\land (x\to y)\equiv x\land y\)
Extends2: Brouwerian algebra

Name: \(n\)-Valued post algebra
Definition: vocabulary \(\langle\lor,\land,',0,1\rangle\)
Page: 29
Chapter: II
Section: 1
Variables: \(n\in\mathbb N\)
Index: 12
Kind: Ex
Definition2: satisfying each identity satisfied by \(\mathbf P_n\)
Conjdef: A theory with

Name: The Post algebra
Definition: vocabulary \(\langle\lor,\land,',0,1\rangle\)
Page: 29
Chapter: II
Section: 1
Variables: \(n\in\mathbb N\)
Notation: \(\mathbf P_n\)
Index: 12
Kind: Ex
Definition2: set \(\{0,\dots,n-1\}\)
Conjdef: The theory
Definition3: with \(0<n-1<n-2<\dots<2<1\)
Definition4: \(i'=i+1\mod n\)

Page: 29
Section: 1
Chapter: II
Axiom: \(c_i0\equiv 0\)
Name: Cylindric algebras of dimension \(n\)
Vocabulary: \(\langle\lor,\land,',0,1,(c_i,(d_{i,j})_{0\le j<n})_{0\le j<n}\rangle\)
Extends: Boolean algebra
Kind: Ex
Index: 13
Axiom2: \(x\le c_ix\)
Axiom3: \(c_i(x\land c_iy)\equiv (c_ix)\land (c_iy)\)
Variables: \(n\in\omega\)
Axiom4: \(c_ic_jx\equiv c_jc_ix\)
Axiom5: \(d_{ii}\equiv 1\)
Axiom6: \(d_{ik}\equiv c_j(d_{ij}\land d_{jk})\) if \(i\ne j\ne k\)
Axiom7: \(c_i(d_{ij}\land x)\land c_i(d_{ij}\land x')\equiv 0\) if \(i\ne j\)

Page: 29
Section: 1
Chapter: II
Axiom: \((x\land y)'\equiv x'\lor y'\), \((x\lor y)'\equiv x'\land y'\)
Name: Ortholattices
Vocabulary: \(\langle\lor,\land,',0,1\rangle\)
Extends: Bounded lattice
Kind: Ex
Index: 14
Axiom2: \((x')'\equiv x\)
Axiom3: \(x\land x'\cong 0, x\lor x'\cong 1\)
Extra: Not boolean because not distributive
Variables: \(n\in\omega\)
Denoted by: \(\mathbf B\)

Page: 29
Section: 1
Chapter: II
Axiom: modular axiom (\(x\le y\implies y\lor (x'\land y)\cong y\))
Name: Orthomodular lattice
Vocabulary: \(\langle\lor,\land,',0,1\rangle\)
Extends: Ortholattices
Kind: Ex
Index: 14
Variables: \(n\in\omega\)

Question: Example of Ortholattice which is not a BA
Page: https://en.wikipedia.org/wiki/Complemented_lattice