Article LATTICE6, MML version 4.99.1005
:: LATTICE6:exreg 1
registration
cluster non empty finite join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr;
end;
:: LATTICE6:condreg 1
registration
cluster non empty finite Lattice-like -> complete (LattStr);
end;
:: LATTICE6:funcnot 1 => LATTICE6:func 1
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of bool the carrier of a1;
func A2 % -> Element of bool the carrier of LattPOSet a1 equals
{b1 % where b1 is Element of the carrier of a1: b1 in a2};
end;
:: LATTICE6:def 1
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of bool the carrier of b1 holds
b2 % = {b3 % where b3 is Element of the carrier of b1: b3 in b2};
:: LATTICE6:funcnot 2 => LATTICE6:func 2
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of bool the carrier of LattPOSet a1;
func % A2 -> Element of bool the carrier of a1 equals
{% b1 where b1 is Element of the carrier of LattPOSet a1: b1 in a2};
end;
:: LATTICE6:def 2
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of bool the carrier of LattPOSet b1 holds
% b2 = {% b3 where b3 is Element of the carrier of LattPOSet b1: b3 in b2};
:: LATTICE6:funcreg 1
registration
let a1 be non empty finite Lattice-like LattStr;
cluster LattPOSet a1 -> strict reflexive transitive antisymmetric well_founded;
end;
:: LATTICE6:attrnot 1 => LATTICE6:attr 1
definition
let a1 be non empty Lattice-like LattStr;
attr a1 is noetherian means
LattPOSet a1 is well_founded;
end;
:: LATTICE6:dfs 3
definiens
let a1 be non empty Lattice-like LattStr;
To prove
a1 is noetherian
it is sufficient to prove
thus LattPOSet a1 is well_founded;
:: LATTICE6:def 3
theorem
for b1 being non empty Lattice-like LattStr holds
b1 is noetherian
iff
LattPOSet b1 is well_founded;
:: LATTICE6:attrnot 2 => LATTICE6:attr 2
definition
let a1 be non empty Lattice-like LattStr;
attr a1 is co-noetherian means
(LattPOSet a1) ~ is well_founded;
end;
:: LATTICE6:dfs 4
definiens
let a1 be non empty Lattice-like LattStr;
To prove
a1 is co-noetherian
it is sufficient to prove
thus (LattPOSet a1) ~ is well_founded;
:: LATTICE6:def 4
theorem
for b1 being non empty Lattice-like LattStr holds
b1 is co-noetherian
iff
(LattPOSet b1) ~ is well_founded;
:: LATTICE6:exreg 2
registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded complete noetherian LattStr;
end;
:: LATTICE6:exreg 3
registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded complete co-noetherian LattStr;
end;
:: LATTICE6:th 1
theorem
for b1 being non empty Lattice-like LattStr holds
b1 is noetherian
iff
b1 .: is co-noetherian;
:: LATTICE6:condreg 2
registration
cluster non empty finite Lattice-like -> noetherian (LattStr);
end;
:: LATTICE6:condreg 3
registration
cluster non empty finite Lattice-like -> co-noetherian (LattStr);
end;
:: LATTICE6:prednot 1 => LATTICE6:pred 1
definition
let a1 be non empty Lattice-like LattStr;
let a2, a3 be Element of the carrier of a1;
pred A2 is-upper-neighbour-of A3 means
a2 <> a3 &
a3 [= a2 &
(for b1 being Element of the carrier of a1
st a3 [= b1 & b1 [= a2 & b1 <> a2
holds b1 = a3);
end;
:: LATTICE6:dfs 5
definiens
let a1 be non empty Lattice-like LattStr;
let a2, a3 be Element of the carrier of a1;
To prove
a2 is-upper-neighbour-of a3
it is sufficient to prove
thus a2 <> a3 &
a3 [= a2 &
(for b1 being Element of the carrier of a1
st a3 [= b1 & b1 [= a2 & b1 <> a2
holds b1 = a3);
:: LATTICE6:def 5
theorem
for b1 being non empty Lattice-like LattStr
for b2, b3 being Element of the carrier of b1 holds
b2 is-upper-neighbour-of b3
iff
b2 <> b3 &
b3 [= b2 &
(for b4 being Element of the carrier of b1
st b3 [= b4 & b4 [= b2 & b4 <> b2
holds b4 = b3);
:: LATTICE6:prednot 2 => LATTICE6:pred 1
notation
let a1 be non empty Lattice-like LattStr;
let a2, a3 be Element of the carrier of a1;
synonym a3 is-lower-neighbour-of a2 for a2 is-upper-neighbour-of a3;
end;
:: LATTICE6:th 2
theorem
for b1 being non empty Lattice-like LattStr
for b2, b3, b4 being Element of the carrier of b1
st b3 <> b4
holds (b3 is-upper-neighbour-of b2 & b4 is-upper-neighbour-of b2 implies b2 = b4 "/\" b3) &
(b2 is-upper-neighbour-of b3 & b2 is-upper-neighbour-of b4 implies b2 = b4 "\/" b3);
:: LATTICE6:th 3
theorem
for b1 being non empty Lattice-like noetherian LattStr
for b2, b3 being Element of the carrier of b1
st b2 [= b3 & b2 <> b3
holds ex b4 being Element of the carrier of b1 st
b4 [= b3 & b4 is-upper-neighbour-of b2;
:: LATTICE6:th 4
theorem
for b1 being non empty Lattice-like co-noetherian LattStr
for b2, b3 being Element of the carrier of b1
st b3 [= b2 & b2 <> b3
holds ex b4 being Element of the carrier of b1 st
b3 [= b4 & b2 is-upper-neighbour-of b4;
:: LATTICE6:th 5
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2 being Element of the carrier of b1 holds
not b2 is-upper-neighbour-of Top b1;
:: LATTICE6:th 6
theorem
for b1 being non empty Lattice-like upper-bounded noetherian LattStr
for b2 being Element of the carrier of b1 holds
b2 = Top b1
iff
for b3 being Element of the carrier of b1 holds
not b3 is-upper-neighbour-of b2;
:: LATTICE6:th 7
theorem
for b1 being non empty Lattice-like lower-bounded LattStr
for b2 being Element of the carrier of b1 holds
not Bottom b1 is-upper-neighbour-of b2;
:: LATTICE6:th 8
theorem
for b1 being non empty Lattice-like lower-bounded co-noetherian LattStr
for b2 being Element of the carrier of b1 holds
b2 = Bottom b1
iff
for b3 being Element of the carrier of b1 holds
not b2 is-upper-neighbour-of b3;
:: LATTICE6:funcnot 3 => LATTICE6:func 3
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
func A2 *' -> Element of the carrier of a1 equals
"/\"({b1 where b1 is Element of the carrier of a1: a2 [= b1 & b1 <> a2},a1);
end;
:: LATTICE6:def 6
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1 holds
b2 *' = "/\"({b3 where b3 is Element of the carrier of b1: b2 [= b3 & b3 <> b2},b1);
:: LATTICE6:funcnot 4 => LATTICE6:func 4
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
func *' A2 -> Element of the carrier of a1 equals
"\/"({b1 where b1 is Element of the carrier of a1: b1 [= a2 & b1 <> a2},a1);
end;
:: LATTICE6:def 7
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1 holds
*' b2 = "\/"({b3 where b3 is Element of the carrier of b1: b3 [= b2 & b3 <> b2},b1);
:: LATTICE6:attrnot 3 => LATTICE6:attr 3
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
attr a2 is completely-meet-irreducible means
a2 *' <> a2;
end;
:: LATTICE6:dfs 8
definiens
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
To prove
a2 is completely-meet-irreducible
it is sufficient to prove
thus a2 *' <> a2;
:: LATTICE6:def 8
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1 holds
b2 is completely-meet-irreducible(b1)
iff
b2 *' <> b2;
:: LATTICE6:attrnot 4 => LATTICE6:attr 4
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
attr a2 is completely-join-irreducible means
*' a2 <> a2;
end;
:: LATTICE6:dfs 9
definiens
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of the carrier of a1;
To prove
a2 is completely-join-irreducible
it is sufficient to prove
thus *' a2 <> a2;
:: LATTICE6:def 9
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1 holds
b2 is completely-join-irreducible(b1)
iff
*' b2 <> b2;
:: LATTICE6:th 9
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1 holds
b2 [= b2 *' & *' b2 [= b2;
:: LATTICE6:th 10
theorem
for b1 being non empty Lattice-like complete LattStr holds
(Top b1) *' = Top b1 & (Top b1) % is meet-irreducible(LattPOSet b1);
:: LATTICE6:th 11
theorem
for b1 being non empty Lattice-like complete LattStr holds
*' Bottom b1 = Bottom b1 & (Bottom b1) % is join-irreducible(LattPOSet b1);
:: LATTICE6:th 12
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is completely-meet-irreducible(b1)
holds b2 *' is-upper-neighbour-of b2 &
(for b3 being Element of the carrier of b1
st b3 is-upper-neighbour-of b2
holds b3 = b2 *');
:: LATTICE6:th 13
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is completely-join-irreducible(b1)
holds b2 is-upper-neighbour-of *' b2 &
(for b3 being Element of the carrier of b1
st b2 is-upper-neighbour-of b3
holds b3 = *' b2);
:: LATTICE6:th 14
theorem
for b1 being non empty Lattice-like complete noetherian LattStr
for b2 being Element of the carrier of b1 holds
b2 is completely-meet-irreducible(b1)
iff
ex b3 being Element of the carrier of b1 st
b3 is-upper-neighbour-of b2 &
(for b4 being Element of the carrier of b1
st b4 is-upper-neighbour-of b2
holds b4 = b3);
:: LATTICE6:th 15
theorem
for b1 being non empty Lattice-like complete co-noetherian LattStr
for b2 being Element of the carrier of b1 holds
b2 is completely-join-irreducible(b1)
iff
ex b3 being Element of the carrier of b1 st
b2 is-upper-neighbour-of b3 &
(for b4 being Element of the carrier of b1
st b2 is-upper-neighbour-of b4
holds b4 = b3);
:: LATTICE6:th 16
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is completely-meet-irreducible(b1)
holds b2 % is meet-irreducible(LattPOSet b1);
:: LATTICE6:th 17
theorem
for b1 being non empty Lattice-like complete noetherian LattStr
for b2 being Element of the carrier of b1
st b2 <> Top b1
holds b2 is completely-meet-irreducible(b1)
iff
b2 % is meet-irreducible(LattPOSet b1);
:: LATTICE6:th 18
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is completely-join-irreducible(b1)
holds b2 % is join-irreducible(LattPOSet b1);
:: LATTICE6:th 19
theorem
for b1 being non empty Lattice-like complete co-noetherian LattStr
for b2 being Element of the carrier of b1
st b2 <> Bottom b1
holds b2 is completely-join-irreducible(b1)
iff
b2 % is join-irreducible(LattPOSet b1);
:: LATTICE6:th 20
theorem
for b1 being non empty finite Lattice-like LattStr
for b2 being Element of the carrier of b1
st b2 <> Bottom b1 & b2 <> Top b1
holds (b2 is completely-meet-irreducible(b1) implies b2 % is meet-irreducible(LattPOSet b1)) &
(b2 % is meet-irreducible(LattPOSet b1) implies b2 is completely-meet-irreducible(b1)) &
(b2 is completely-join-irreducible(b1) implies b2 % is join-irreducible(LattPOSet b1)) &
(b2 % is join-irreducible(LattPOSet b1) implies b2 is completely-join-irreducible(b1));
:: LATTICE6:attrnot 5 => LATTICE6:attr 5
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of the carrier of a1;
attr a2 is atomic means
a2 is-upper-neighbour-of Bottom a1;
end;
:: LATTICE6:dfs 10
definiens
let a1 be non empty Lattice-like LattStr;
let a2 be Element of the carrier of a1;
To prove
a2 is atomic
it is sufficient to prove
thus a2 is-upper-neighbour-of Bottom a1;
:: LATTICE6:def 10
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1 holds
b2 is atomic(b1)
iff
b2 is-upper-neighbour-of Bottom b1;
:: LATTICE6:attrnot 6 => LATTICE6:attr 6
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of the carrier of a1;
attr a2 is co-atomic means
Top a1 is-upper-neighbour-of a2;
end;
:: LATTICE6:dfs 11
definiens
let a1 be non empty Lattice-like LattStr;
let a2 be Element of the carrier of a1;
To prove
a2 is co-atomic
it is sufficient to prove
thus Top a1 is-upper-neighbour-of a2;
:: LATTICE6:def 11
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1 holds
b2 is co-atomic(b1)
iff
Top b1 is-upper-neighbour-of b2;
:: LATTICE6:th 21
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is atomic(b1)
holds b2 is completely-join-irreducible(b1);
:: LATTICE6:th 22
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of the carrier of b1
st b2 is co-atomic(b1)
holds b2 is completely-meet-irreducible(b1);
:: LATTICE6:attrnot 7 => LATTICE6:attr 7
definition
let a1 be non empty Lattice-like LattStr;
attr a1 is atomic means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool the carrier of a1 st
(for b3 being Element of the carrier of a1
st b3 in b2
holds b3 is atomic(a1)) &
b1 = "\/"(b2,a1);
end;
:: LATTICE6:dfs 12
definiens
let a1 be non empty Lattice-like LattStr;
To prove
a1 is atomic
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool the carrier of a1 st
(for b3 being Element of the carrier of a1
st b3 in b2
holds b3 is atomic(a1)) &
b1 = "\/"(b2,a1);
:: LATTICE6:def 12
theorem
for b1 being non empty Lattice-like LattStr holds
b1 is atomic
iff
for b2 being Element of the carrier of b1 holds
ex b3 being Element of bool the carrier of b1 st
(for b4 being Element of the carrier of b1
st b4 in b3
holds b4 is atomic(b1)) &
b2 = "\/"(b3,b1);
:: LATTICE6:exreg 4
registration
cluster non empty trivial strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr;
end;
:: LATTICE6:exreg 5
registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like complete atomic LattStr;
end;
:: LATTICE6:attrnot 8 => LATTICE6:attr 8
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of bool the carrier of a1;
attr a2 is supremum-dense means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool a2 st
b1 = "\/"(b2,a1);
end;
:: LATTICE6:dfs 13
definiens
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is supremum-dense
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool a2 st
b1 = "\/"(b2,a1);
:: LATTICE6:def 13
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is supremum-dense(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being Element of bool b2 st
b3 = "\/"(b4,b1);
:: LATTICE6:attrnot 9 => LATTICE6:attr 9
definition
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of bool the carrier of a1;
attr a2 is infimum-dense means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool a2 st
b1 = "/\"(b2,a1);
end;
:: LATTICE6:dfs 14
definiens
let a1 be non empty Lattice-like complete LattStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is infimum-dense
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool a2 st
b1 = "/\"(b2,a1);
:: LATTICE6:def 14
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is infimum-dense(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being Element of bool b2 st
b3 = "/\"(b4,b1);
:: LATTICE6:th 23
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is supremum-dense(b1)
iff
for b3 being Element of the carrier of b1 holds
b3 = "\/"({b4 where b4 is Element of the carrier of b1: b4 in b2 & b4 [= b3},b1);
:: LATTICE6:th 24
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is infimum-dense(b1)
iff
for b3 being Element of the carrier of b1 holds
b3 = "/\"({b4 where b4 is Element of the carrier of b1: b4 in b2 & b3 [= b4},b1);
:: LATTICE6:th 25
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is infimum-dense(b1)
iff
b2 % is order-generating(LattPOSet b1);
:: LATTICE6:funcnot 5 => LATTICE6:func 5
definition
let a1 be non empty Lattice-like complete LattStr;
func MIRRS A1 -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: b1 is completely-meet-irreducible(a1)};
end;
:: LATTICE6:def 15
theorem
for b1 being non empty Lattice-like complete LattStr holds
MIRRS b1 = {b2 where b2 is Element of the carrier of b1: b2 is completely-meet-irreducible(b1)};
:: LATTICE6:funcnot 6 => LATTICE6:func 6
definition
let a1 be non empty Lattice-like complete LattStr;
func JIRRS A1 -> Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: b1 is completely-join-irreducible(a1)};
end;
:: LATTICE6:def 16
theorem
for b1 being non empty Lattice-like complete LattStr holds
JIRRS b1 = {b2 where b2 is Element of the carrier of b1: b2 is completely-join-irreducible(b1)};
:: LATTICE6:th 26
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1
st b2 is supremum-dense(b1)
holds JIRRS b1 c= b2;
:: LATTICE6:th 27
theorem
for b1 being non empty Lattice-like complete LattStr
for b2 being Element of bool the carrier of b1
st b2 is infimum-dense(b1)
holds MIRRS b1 c= b2;
:: LATTICE6:funcreg 2
registration
let a1 be non empty Lattice-like complete co-noetherian LattStr;
cluster MIRRS a1 -> infimum-dense;
end;
:: LATTICE6:funcreg 3
registration
let a1 be non empty Lattice-like complete noetherian LattStr;
cluster JIRRS a1 -> supremum-dense;
end;