Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Introduction to Lattice Theory

**by****Stanislaw Zukowski**- Received April 14, 1989
- MML identifier: LATTICES

- [ Mizar article, MML identifier index ]

environ vocabulary BINOP_1, BOOLE, FINSUB_1, FUNCT_1, SUBSET_1, LATTICES; notation XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, BINOP_1, FINSUB_1; constructors STRUCT_0, BINOP_1, FINSUB_1, XBOOLE_0; clusters FINSUB_1, STRUCT_0, SUBSET_1, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin definition struct(1-sorted) /\-SemiLattStr (# carrier -> set, L_meet -> BinOp of the carrier #); end; definition struct(1-sorted) \/-SemiLattStr (# carrier -> set, L_join -> BinOp of the carrier #); end; definition struct(/\-SemiLattStr,\/-SemiLattStr) LattStr (# carrier -> set, L_join, L_meet -> BinOp of the carrier #); end; definition cluster strict non empty \/-SemiLattStr; cluster strict non empty /\-SemiLattStr; cluster strict non empty LattStr; end; definition let G be non empty \/-SemiLattStr, p, q be Element of G; func p"\/"q -> Element of G equals :: LATTICES:def 1 (the L_join of G).(p,q); end; definition let G be non empty /\-SemiLattStr, p, q be Element of G; func p"/\"q -> Element of G equals :: LATTICES:def 2 (the L_meet of G).(p,q); end; definition let G be non empty \/-SemiLattStr, p, q be Element of G; pred p [= q means :: LATTICES:def 3 p"\/"q = q; end; definition let IT be non empty \/-SemiLattStr; attr IT is join-commutative means :: LATTICES:def 4 for a,b being Element of IT holds a"\/"b = b"\/"a; attr IT is join-associative means :: LATTICES:def 5 for a,b,c being Element of IT holds a"\/"(b"\/"c) = (a"\/"b) "\/"c; end; definition let IT be non empty /\-SemiLattStr; attr IT is meet-commutative means :: LATTICES:def 6 for a,b being Element of IT holds a"/\"b = b"/\"a; attr IT is meet-associative means :: LATTICES:def 7 for a,b,c being Element of IT holds a"/\"(b"/\"c) = (a"/\"b) "/\"c; end; definition let IT be non empty LattStr; attr IT is meet-absorbing means :: LATTICES:def 8 for a,b being Element of IT holds (a"/\"b)"\/"b = b; attr IT is join-absorbing means :: LATTICES:def 9 for a,b being Element of IT holds a"/\"(a"\/"b)=a; end; definition let IT be non empty LattStr; attr IT is Lattice-like means :: LATTICES:def 10 IT is join-commutative join-associative meet-absorbing meet-commutative meet-associative join-absorbing; end; definition cluster Lattice-like -> join-commutative join-associative meet-absorbing meet-commutative meet-associative join-absorbing (non empty LattStr); cluster join-commutative join-associative meet-absorbing meet-commutative meet-associative join-absorbing -> Lattice-like (non empty LattStr); end; definition cluster strict join-commutative join-associative (non empty \/-SemiLattStr); cluster strict meet-commutative meet-associative (non empty /\-SemiLattStr); cluster strict Lattice-like (non empty LattStr); end; definition mode Lattice is Lattice-like (non empty LattStr); end; definition let L be join-commutative (non empty \/-SemiLattStr), a, b be Element of L; redefine func a"\/"b; commutativity; end; definition let L be meet-commutative (non empty /\-SemiLattStr), a, b be Element of L; redefine func a"/\"b; commutativity; end; definition let IT be non empty LattStr; attr IT is distributive means :: LATTICES:def 11 for a,b,c being Element of IT holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c); end; definition let IT be non empty LattStr; attr IT is modular means :: LATTICES:def 12 for a,b,c being Element of IT st a [= c holds a"\/"(b"/\"c) = (a"\/"b)"/\"c; end; definition let IT be non empty /\-SemiLattStr; attr IT is lower-bounded means :: LATTICES:def 13 ex c being Element of IT st for a being Element of IT holds c"/\"a = c & a"/\"c = c; end; definition let IT be non empty \/-SemiLattStr; attr IT is upper-bounded means :: LATTICES:def 14 ex c being Element of IT st for a being Element of IT holds c"\/"a = c & a"\/"c = c; end; definition cluster strict distributive lower-bounded upper-bounded modular Lattice; end; definition mode D_Lattice is distributive Lattice; mode M_Lattice is modular Lattice; mode 0_Lattice is lower-bounded Lattice; mode 1_Lattice is upper-bounded Lattice; end; definition let IT be non empty LattStr; attr IT is bounded means :: LATTICES:def 15 IT is lower-bounded upper-bounded; end; definition cluster lower-bounded upper-bounded -> bounded (non empty LattStr); cluster bounded -> lower-bounded upper-bounded (non empty LattStr); end; definition cluster bounded strict Lattice; end; definition mode 01_Lattice is bounded Lattice; end; definition let L be non empty /\-SemiLattStr; assume L is lower-bounded; func Bottom L -> Element of L means :: LATTICES:def 16 for a being Element of L holds it "/\" a = it & a "/\" it = it; end; definition let L be non empty \/-SemiLattStr; assume L is upper-bounded; func Top L -> Element of L means :: LATTICES:def 17 for a being Element of L holds it "\/" a = it & a "\/" it = it; end; definition let L be non empty LattStr, a, b be Element of L; pred a is_a_complement_of b means :: LATTICES:def 18 a"\/"b = Top L & b"\/"a = Top L & a"/\"b = Bottom L & b"/\"a = Bottom L; end; definition let IT be non empty LattStr; attr IT is complemented means :: LATTICES:def 19 for b being Element of IT ex a being Element of IT st a is_a_complement_of b; end; definition cluster bounded complemented strict Lattice; end; definition mode C_Lattice is complemented 01_Lattice; end; definition let IT be non empty LattStr; attr IT is Boolean means :: LATTICES:def 20 IT is bounded complemented distributive; end; definition cluster Boolean -> bounded complemented distributive (non empty LattStr); cluster bounded complemented distributive -> Boolean (non empty LattStr); end; definition cluster Boolean strict Lattice; end; definition mode B_Lattice is Boolean Lattice; end; reserve L for meet-absorbing join-absorbing meet-commutative (non empty LattStr); reserve a for Element of L; canceled 16; theorem :: LATTICES:17 a"\/"a = a; theorem :: LATTICES:18 a"/\"a = a; reserve L for Lattice; reserve a, b, c for Element of L; theorem :: LATTICES:19 (for a,b,c holds a"/\"(b"\/"c) = (a"/\"b)"\/"(a"/\"c)) iff (for a,b,c holds a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c)); canceled; theorem :: LATTICES:21 for L being meet-absorbing join-absorbing (non empty LattStr), a, b being Element of L holds a [= b iff a"/\"b = a; theorem :: LATTICES:22 for L being meet-absorbing join-absorbing join-associative meet-commutative (non empty LattStr), a, b being Element of L holds a [= a"\/"b; theorem :: LATTICES:23 for L being meet-absorbing meet-commutative (non empty LattStr), a, b being Element of L holds a"/\"b [= a; definition let L be meet-absorbing join-absorbing meet-commutative (non empty LattStr), a, b be Element of L; redefine pred a [= b; reflexivity; end; canceled; theorem :: LATTICES:25 for L being join-associative (non empty \/-SemiLattStr), a, b, c being Element of L holds a [= b & b [= c implies a [= c; theorem :: LATTICES:26 for L being join-commutative (non empty \/-SemiLattStr), a, b being Element of L holds a [= b & b [= a implies a=b; theorem :: LATTICES:27 for L being meet-absorbing join-absorbing meet-associative (non empty LattStr), a, b, c being Element of L holds a [= b implies a"/\"c [= b"/\"c; canceled; theorem :: LATTICES:29 for L being Lattice holds (for a,b,c being Element of L holds (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a) = (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a)) implies L is distributive; reserve L for D_Lattice; reserve a, b, c for Element of L; canceled; theorem :: LATTICES:31 a"\/"(b"/\"c) = (a"\/"b)"/\"(a"\/"c); theorem :: LATTICES:32 c"/\"a = c"/\"b & c"\/"a = c"\/"b implies a=b; canceled; theorem :: LATTICES:34 (a"\/"b)"/\"(b"\/"c)"/\"(c"\/"a) = (a"/\"b)"\/"(b"/\"c)"\/"(c"/\"a); definition cluster distributive -> modular Lattice; end; reserve L for 0_Lattice; reserve a for Element of L; canceled 4; theorem :: LATTICES:39 Bottom L"\/"a = a; theorem :: LATTICES:40 Bottom L"/\"a = Bottom L; theorem :: LATTICES:41 Bottom L [= a; reserve L for 1_Lattice; reserve a for Element of L; canceled; theorem :: LATTICES:43 Top L"/\"a = a; theorem :: LATTICES:44 Top L"\/"a = Top L; theorem :: LATTICES:45 a [= Top L; definition let L be non empty LattStr, x be Element of L; assume L is complemented D_Lattice; func x` -> Element of L means :: LATTICES:def 21 it is_a_complement_of x; end; reserve L for B_Lattice; reserve a, b for Element of L; canceled; theorem :: LATTICES:47 a`"/\"a = Bottom L; theorem :: LATTICES:48 a`"\/"a = Top L; theorem :: LATTICES:49 a`` = a; theorem :: LATTICES:50 ( a"/\"b )` = a`"\/" b`; theorem :: LATTICES:51 ( a"\/"b )` = a`"/\" b`; theorem :: LATTICES:52 b"/\"a = Bottom L iff b [= a`; theorem :: LATTICES:53 a [= b implies b` [= a`;

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