Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

### The abstract of the Mizar article:

### On the Category of Posets

**by****Adam Grabowski**- Received January 22, 1996
- MML identifier: ORDERS_3

- [ Mizar article, MML identifier index ]

environ vocabulary ORDERS_1, NATTRA_1, RELAT_1, RELAT_2, BOOLE, WELLORD1, SUBSET_1, PRE_TOPC, SEQM_3, FUNCT_1, FUNCT_2, UNIALG_3, FRAENKEL, CAT_5, CAT_1, ALTCAT_1, PBOOLE, PRALG_1, BINOP_1, ORDERS_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELAT_2, FUNCT_1, CAT_5, STRUCT_0, MCART_1, WELLORD1, PARTFUN1, BINOP_1, MULTOP_1, RELSET_1, PRE_TOPC, ORDERS_1, CAT_1, ENS_1, FRAENKEL, PBOOLE, GRCAT_1, FUNCT_2, PRALG_1, ALTCAT_1; constructors RELAT_2, ORDERS_1, WELLORD1, ALTCAT_1, ENS_1, CAT_5, DOMAIN_1, TOPS_2, GRCAT_1; clusters SUBSET_1, RELSET_1, ORDERS_1, STRUCT_0, FUNCT_1, FRAENKEL, CAT_5, ALTCAT_1, ENS_1, PARTFUN1, XBOOLE_0; requirements BOOLE, SUBSET; begin :: Preliminaries definition let IT be RelStr; attr IT is discrete means :: ORDERS_3:def 1 the InternalRel of IT = id (the carrier of IT); end; definition cluster strict discrete non empty Poset; cluster strict discrete empty Poset; end; definition cluster RelStr (#{},id {}#) -> empty; let P be empty RelStr; cluster the InternalRel of P -> empty; end; definition cluster empty -> discrete RelStr; end; definition let P be RelStr; let IT be Subset of P; attr IT is disconnected means :: ORDERS_3:def 2 ex A,B be Subset of P st A <> {} & B <> {} & IT = A \/ B & A misses B & the InternalRel of P = (the InternalRel of P) |_2 A \/ (the InternalRel of P) |_2 B; antonym IT is connected; end; definition let IT be RelStr; attr IT is disconnected means :: ORDERS_3:def 3 [#] IT is disconnected; antonym IT is connected; end; reserve T for non empty RelStr, a for Element of T; theorem :: ORDERS_3:1 for DP be discrete non empty RelStr, x,y be Element of DP holds x <= y iff x = y; theorem :: ORDERS_3:2 for R be Relation, a be set st R is Order of {a} holds R = id {a}; theorem :: ORDERS_3:3 T is reflexive & [#] T = {a} implies T is discrete; reserve a for set; theorem :: ORDERS_3:4 [#] T = {a} implies T is connected; theorem :: ORDERS_3:5 for DP be discrete non empty Poset st (ex a,b be Element of DP st a <> b) holds DP is disconnected; definition cluster strict connected (non empty Poset); cluster strict disconnected discrete (non empty Poset); end; begin :: Category of Posets definition let IT be set; attr IT is POSet_set-like means :: ORDERS_3:def 4 for a be set st a in IT holds a is non empty Poset; end; definition cluster non empty POSet_set-like set; end; definition mode POSet_set is POSet_set-like set; end; definition let P be non empty POSet_set; redefine mode Element of P -> non empty Poset; end; definition let L1,L2 be RelStr; let f be map of L1, L2; attr f is monotone means :: ORDERS_3:def 5 for x,y being Element of L1 st x <= y for a,b being Element of L2 st a = f.x & b = f.y holds a <= b; end; reserve P for non empty POSet_set, A,B for Element of P; definition let A,B be RelStr; func MonFuncs (A,B) means :: ORDERS_3:def 6 a in it iff ex f be map of A, B st a = f & f in Funcs (the carrier of A, the carrier of B) & f is monotone; end; theorem :: ORDERS_3:6 for A,B,C be non empty RelStr for f,g be Function st f in MonFuncs (A,B) & g in MonFuncs (B,C) holds (g*f) in MonFuncs (A,C); theorem :: ORDERS_3:7 id the carrier of T in MonFuncs (T,T); definition let T; cluster MonFuncs (T,T) -> non empty; end; definition let X be set; func Carr X -> set means :: ORDERS_3:def 7 a in it iff ex s be 1-sorted st s in X & a = the carrier of s; end; definition let P; cluster Carr P -> non empty; end; theorem :: ORDERS_3:8 for f be 1-sorted holds Carr {f} = {the carrier of f}; theorem :: ORDERS_3:9 for f,g be 1-sorted holds Carr {f,g} = {the carrier of f, the carrier of g} ; theorem :: ORDERS_3:10 MonFuncs (A,B) c= Funcs Carr P; theorem :: ORDERS_3:11 for A,B be RelStr holds MonFuncs (A,B) c= Funcs (the carrier of A,the carrier of B); definition let A,B be non empty Poset; cluster MonFuncs (A,B) -> functional; end; definition let P be non empty POSet_set; func POSCat P -> strict with_triple-like_morphisms Category means :: ORDERS_3:def 8 the Objects of it = P & (for a,b be Element of P, f be Element of Funcs Carr P st f in MonFuncs (a,b) holds [[a,b],f] is Morphism of it) & (for m be Morphism of it ex a,b be Element of P, f be Element of Funcs (Carr P) st m = [[a,b],f] & f in MonFuncs (a,b)) & for m1,m2 be (Morphism of it), a1,a2,a3 be Element of P, f1,f2 be Element of Funcs (Carr P) st m1 = [[a1,a2],f1] & m2 = [[a2,a3],f2] holds m2*m1 = [[a1,a3], f2*f1]; end; begin :: Alternative Category of Posets scheme AltCatEx {A() -> non empty set, F(set,set) -> functional set }: ex C be strict AltCatStr st the carrier of C = A() & for i,j be Element of A() holds (the Arrows of C).(i,j) = F (i,j) & for i,j,k be Element of A() holds (the Comp of C).(i,j,k) = FuncComp ( F(i,j) , F(j,k) ) provided for i,j,k be Element of A() for f,g be Function st f in F(i,j) & g in F(j,k) holds g * f in F (i,k); scheme AltCatUniq {A() -> non empty set, F(set,set) -> functional set } : for C1,C2 be strict AltCatStr st ( the carrier of C1 = A() & for i,j be Element of A() holds (the Arrows of C1).(i,j) = F (i,j) & for i,j,k be Element of A() holds (the Comp of C1).(i,j,k) = FuncComp ( F(i,j) , F(j,k) ) ) & ( the carrier of C2 = A() & for i,j be Element of A() holds (the Arrows of C2).(i,j) = F (i,j) & for i,j,k be Element of A() holds (the Comp of C2).(i,j,k) = FuncComp ( F(i,j) , F(j,k) ) ) holds C1 = C2; definition let P be non empty POSet_set; func POSAltCat P -> strict AltCatStr means :: ORDERS_3:def 9 the carrier of it = P & for i,j be Element of P holds (the Arrows of it).(i,j) = MonFuncs (i,j) & for i,j,k be Element of P holds (the Comp of it).(i,j,k) = FuncComp ( MonFuncs (i,j) , MonFuncs (j,k) ); end; definition let P be non empty POSet_set; cluster POSAltCat P -> transitive non empty; end; definition let P be non empty POSet_set; cluster POSAltCat P -> associative with_units; end; theorem :: ORDERS_3:12 for o1,o2 be object of POSAltCat P, A,B be Element of P st o1 = A & o2 = B holds <^ o1 , o2 ^> c= Funcs (the carrier of A, the carrier of B);

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