Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories

Transcription

1 ppl teor Struct (2008) 6: OI 0.007/s onstruction nd Properties o dhesive nd Wek dhesive Hih-Level Replcement teories Ulrike Prne Hrtmut Ehri Leen Lmers Received: 8 July 2006 / ccepted: 22 uust 2007 / Pulished online: 3 Octoer 2007 Spriner Science + usiness Medi.V strct s presented in Ehri et l. (Fundmentls o leric Grph Trnsormtion, ETS Monorphs, Spriner, 2006), dhesive hih-level replcement (HLR) cteories nd systems re n dequte rmework or severl kinds o trnsormtion systems sed on the doule pushout pproch. Since (wek) dhesive HLR cteories re closed under product, slice, coslice, comm nd unctor cteory constructions, it is possile to uild new (wek) dhesive HLR cteories rom existin ones. ut or the enerl results o trnsormtion systems, s dditionl properties initil pushouts, inry coproducts comptile with specil morphism clss M nd pir ctoriztion re needed to otin the ull theory. In this pper, we nlyze under which conditions these dditionl properties re preserved y the cteoricl constructions in order to void checkin these properties explicitly. Keywords dhesive HLR cteories Initil pushouts oproducts Pir ctoriztion Mthemtics Suject lssiictions (2000) U. Prne () H. Ehri L. Lmers Technicl University o erlin, Sekr. FR6-, Frnklinstr. 28/29, 0587 erlin, Germny e-mil: uprne@cs.tu-erlin.de H. Ehri e-mil: ehri@cs.tu-erlin.de L. Lmers e-mil: leen@cs.tu-erlin.de

2 366 U. Prne et l. Introduction dhesive hih-level replcement cteories nd systems hve een introduced recently in [6] s n intertion o the concepts o hih-level replcement (HLR) systems in [4, 5] enerlizin rph trnsormtion systems nd o dhesive cteories in [, 2] enerlizin isimultion conruences. In [3], dhesive HLR systems re shown to e n dequte uniyin rmework or severl interestin kinds o rph nd net trnsormtion systems. The min ide is to enerlize the leric pproch o rph trnsormtions introduced in [2, 7] rom rphs to hih-level structures nd to instntite them with vrious kinds o rphs, Petri nets, leric speciictions, nd typed ttriuted rphs. dhesive nd wek dhesive HLR cteories re the sis o dhesive HLR systems. The concept o these cteories is sed on the existence nd comptiility o suitle pushouts nd pullcks, which re essentil or the so-clled vn Kmpen (VK) squres, which hve een introduced or dhesive cteories in []. The ide o VK squre is tht o pushout which is stle under pullcks, nd, vice vers, tht pullcks re stle under comined pushouts nd pullcks. The nme vn Kmpen is derived rom the reltionship etween these squres nd the Vn Kmpen theorem in topoloy (see []). While dhesive cteories re sed on the clss o ll monomorphisms, dhesive nd wek dhesive HLR cteories (, M) re sed on suitle suclss M o monomorphisms. This more lexile clss M is essentil or some importnt exmples, such s typed ttriuted rphs, to ecome n dhesive HLR cteory. The concept o wek dhesive HLR cteories is lso importnt, ecuse some other exmples, such s plce/trnsition nets nd leric hih-level nets (see [8]) stisy only weker version o dhesive HLR cteories. This weker version, however, is still suicient to otin the sic min results or dhesive HLR systems in [3]. In [], it ws lredy oserved how to extend the construction o dhesive cteories rom sic exmples such s the cteory Sets o sets nd unctions to more complex exmples such s the cteory Grphs o rphs nd rph morphisms nd the cteory Grphs TG o rphs nd rph morphisms typed over type rph TG. In ct, it is climed in [] tht dhesive cteories re closed under product, slice, coslice nd unctor cteory constructions. This hs een extended in [6] to dhesive HLR cteories nd in [3] it is shown in ddition tht dhesive nd lso wek dhesive HLR cteories re closed under comm cteory constructions. In [6] it is shown lredy tht some results in the theory o dhesive HLR systems require some dditionl properties like inite coproducts comptile with M, initil pushouts nd pir ctoriztion. Especilly the existence nd construction o initil pushouts is nontrivil to e shown in severl exmple cteories. For this reson we study in this pper how r the constructions o (wek) dhesive HLR cteories discussed ove llow to preserve lso these dditionl properties in order to void checkin these properties explicitly. This pper is ornized s ollows. In Section 2, we introduce (wek) dhesive HLR cteories s presented in [3] nd cite the importnt onstruction theorem. In Section 3, we nlyze underwhich conditions the dditionl propertiesrepreserved under the cteoricl constructions mentioned ove. Section 4 ives conclusion nd n overview o uture work.

3 onstruction nd properties o dhesive HLR cteories dhesive nd Wek dhesive HLR teories In this section, we introduce the notion o dhesive nd wek dhesive HLR cteories, summrized s (wek) dhesive HLR cteories, nd present their closure under certin cteoricl constructions. The reder is ssumed to e milir with the sic notions o cteory theory, s presented in, e.., [3, 9, 3]. For more motivtion nd exmples we reer to [3]. The sic notion o (wek) dhesive HLR cteories re vn Kmpen (VK) squres. The ide o VK squre is tht o pushout (PO) which is stle under pullcks, nd, vice vers, tht pullcks re stle under comined pushouts nd pullcks. einition ([Wek] Vn Kmpen Squre). pushout () is vn Kmpen squre i, or ny commuttive cue (2) with () in the ottom nd where the ck ces re pullcks, the ollowin sttement holds: the top ce is pushout i the ront ces re pullcks: m () n c n n d m m (2) Given morphism clss M, pushout () with m M is wek VK squre i the ove property holds or ll commuttive cues with M or, c, d M. Remrk Given morphism clss M, pushout () is pushout lon M-morphisms i m M (or M). nloously, pullck () is pullck lon M-morphisms i n M (or M). Exmple In the ollowin dirm VK squre lon n injective unction in Sets is shown on the let-hnd side. ll morphisms re inclusions, except o 0 nd re mpped to nd 2 nd 3 to 2. ritrry pushouts re stle under pullcks in Sets. This mens tht one direction o the VK squre property is lso vlid or ritrry morphisms. However, the other direction is not necessrily vlid. The cue on the riht-hnd side is such counterexmple, or ritrry unctions: ll ces commute, the ottom nd the top re pushouts, nd the ck ces re pullcks. ut, oviously, the ront ces re not pullcks, nd thereore the pushout in the ottom ils to e VK squre.

4 368 U. Prne et l. { } {0, } {0, } π {0, } {0, } π 2 { } {, 2, 3} {0, } {0,, 2, 3} { } { } + mod2 {0, } {0, } {, 2} {0,, 2} { } { } einition 2 ([Wek] dhesivehlr teory). cteory with morphism clss M is clled (wek) dhesive HLR cteory i:. M is clss o monomorphisms closed under isomorphisms, composition ( : M, : M M), nd decomposition ( M, M M). 2. hs pushouts nd pullcks lon M-morphisms, nd M-morphisms re closed under pushouts nd pullcks. 3. Pushouts in lon M-morphisms re (wek) VK squres. Exmple 2 The cteories Sets nd Grphs o sets nd rphs, respectively, toether with the clss M o injective morphisms re dhesive HLR cteories. For the typin o rphs, distinuished rph TG, clled type rph, deines the ville node nd ede types. Then, typed rph is rph G toether with typin morphism t : G TG. Typed rph morphisms re rph morphisms tht preserve the typin. The cteory Grphs TG o typed rph toether with the morphism clss M o injective morphisms is n dhesive HLR cteory. lso the cteory Grphs TG o typed ttriuted rphs with the clss M o injective morphisms with isomorphic dt prt is n dhesive HLR cteory. For typed ttriuted rphs, we hve typin s well s ttriutes or nodes nd edes, where the ttriute vlues re speciied y some ler. Typed ttriuted rph morphisms mp the rphs nd the lers such tht oth types nd ttriution is preserved. The cteory PTNets o Petri nets with the morphism clss M o injective morphisms is not n dhesive HLR cteory, ut wek dhesive HLR cteory, since only pullcks lon M-morphisms re constructed componentwise, ut not over enerl morphisms (see [8]). (Wek) dhesive HLR cteories re closed under product, slice, coslice, unctor, nd comm cteory constructions, where some o these notions re explined in the remrk elow. This mens tht we cn construct new (wek) dhesive HLR cteories rom iven ones.

5 onstruction nd properties o dhesive HLR cteories 369 Theorem (onstruction o [Wek] dhesive HLR teories). I (, M ) nd (, M 2 ) re (wek) dhesive HLR cteories, then we hve the ollowin results.. The product cteory (, M M 2 ) is (wek) dhesive HLR cteory. 2. Theslicecteory(\X, M \X) is (wek) dhesive HLR cteory or ny oject X in. 3. The coslice cteory (X\, M X\) is (wek) dhesive HLR cteory or ny oject X in. 4. The comm cteory (omt(f, G; I), M) with M = (M M 2 ) Mor omt(f,g;i) nd with unctors F :, G: is (wek) dhesive HLR cteory, i F preserves pushouts lon M -morphisms nd G preserves pullcks (lon M 2 -morphisms). 5. The unctor cteory ([X, ], M nturl trns ormtions) is (wek) dhesive HLR cteory or every cteory X. Remrk Given unctors F : nd G :, nd n index set I, the ojects o the comm cteory (omt(f, G; I) re tuples (,,(op i ) i I ) with, nd op i : F() G(). morphism : (,, op i ) (,, op i ) consists o morphisms : in nd : in such tht G( ) op i = op i F( ) or ll i I. F() F( ) op i G() G( ) F( ) G( ) op i In unctor cteory [X, ], nm -nturl trnsormtion is nturl trnsormtion t : F G, where ll morphisms t X : F(X) G(X) re in M. Proo See [3]. Exmple 3. The cteory Grphs Grphs s the product cteory over Grphs toether with injective morphisms is n dhesive HLR cteory (see item 5). 2. The cteory Grphs TG s the slice cteory Grphs\TG toether with injective morphisms is n dhesive HLR cteory. 3. The cteory PSets o sets nd prtil unctions is isomorphic to the coslice cteory {}\Sets nd thus PSets toether with injective totl unctions is n dhesive HLR cteory. 4. The cteory PTNets o Petri nets is isomorphic to the comm cteory omt(f, G; I) with F = Id : Sets Sets, G = : Sets Sets nd I = {pre, post},where is the ree commuttive monoid over.sincef preserves pushouts nd G preserves pullcks lon injective morphisms (ut does not preserveenerlpullcks), PTNets toether with injective Petri net morphisms is wek dhesive HLR cteory.

6 370 U. Prne et l. 5. The cteory Grphs is isomorphic to the unctor cteory [S 2, Sets] with S 2 = nd thus, toether with injective rph morphisms, n dhesive HLR cteory. To simpliy the proos, product, slice nd coslice cteories cn e seen s specil cses o comm cteories. Hence, some results or these cteories cn e otined rom the correspondin results or comm cteories. Fct (Product, Slice nd oslice teory s omm teory). For product, slice nd coslice cteories, we hve the ollowin isomorphic comm cteories:. = omt(! :,! :, ), 2. \X = omt(id :, X :, {}) nd 3. X\ = omt(x :, id :, {}), where is the inl cteory,! : is the inl morphism rom, ndx: mps to X. In the ollowin, we need the ct elow out pushouts in comm cteories. It shows, tht pushouts lon M-morphisms in comm cteories sed on (wek) dhesive HLR cteories cn e uild componentwise rom the underlyin cteories. Fct 2 (omponentwise Pushouts in omm teories). Given wek dhesive HLR cteories (, M ) nd (, M 2 ) nd unctors F : nd G :, where F preserves pushouts lon M -morphisms, tht led to comm cteory omt(f, G; I). Then or ojects = (, 2, op i ), = (, 2, op i ), = (, 2, op i ) omt(f, G; I) nd morphisms = (, 2 ) :, = (, 2 ) : with M M 2 we hve: The dirm () is pushout in omt(f, G; I) i () nd () re pushouts in nd, respectively, with = (, 2 ) nd c = (c, c 2 ) () () 2 2 () c 2 c 2 2 c Proo Given the morphisms nd in (), nd the pushouts () nd () in nd, respectively. We hve to show tht () is pushout in omt(f, G; I). Since F preserves pushouts lon M -morphisms, with M the dirm (2) is pushout. Then = (, 2, op i ) is n oject in omt(f, G; I), where, or i I, op i is the y (2) nd G(c 2 ) op i F( ) = G(c 2 ) G( 2 ) op i = G( 2 ) G( 2 ) op i = G( 2 ) op i F( ) induced morphism with op i F(c ) = G(c 2 ) op i nd op i F( ) = G( 2 ) op i. Thereore c = (c, c 2 ) nd = (, 2 ) re morphisms in omt(f, G; I) such tht () commutes.

7 onstruction nd properties o dhesive HLR cteories 37 G( 2 ) G( 2 ) G( 2 ) op i op i F( ) F( ) F( ) (2) G( 2 ) F( ) F( ) G( 2 ) F( ) F(c ) F( ) F(k ) G(k 2 ) op i op i F(x ) G( 2 ) G(c 2 ) G( 2 ) F(h ) F(X ) G(h 2 ) G(x 2 ) op X i G(X 2 ) It remins to show tht () is pushout. Given n oject X = (X, X 2, opi X ) nd morphisms h = (h, h 2 ) : X nd k = (k, k 2 ) : X in omt(f, G; I) such tht h = k. From pushouts () nd () we otin unique morphisms x : X nd x 2 : 2 X 2 such tht x i c i = h i nd x i i = k i or i =, 2. Since (2) is pushout, rom G(x 2 ) opi F(c ) = G(x 2 ) G(c 2 ) opi = G(h 2 ) opi = opi X F(h ) = opi X F(x ) F(c ) nd G(x 2 ) opi F( ) = G(x 2 ) G( 2 ) opi = G(k 2 ) opi = opi X F(k ) = opi X F(x ) F( ) it ollows tht G(x 2 ) opi = opi X F(x ). Thereore x = (x, x 2 ) omt(f, G; I),ndxis unique with respect to x c = h nd x = k. Given the pushout () in omt(f, G; I), we hve to show tht () nd () re pushouts in nd, respectively. Since (, M ) nd (, M 2 ) re wek dhesive HLR cteories there exist pushouts ( ) nd ( ) over nd M in nd over 2 nd 2 M 2 in, respectively ( ) ( ) 2 2 ( ) e E 2 e 2 E 2 e E Thereore (usin ) there is correspondin pushout ( ) in omt(f, G; I) over nd with E = (E, E 2, opi E ), e = (e, e 2 ) nd = (, 2 ). Since pushouts re unique up to isomorphism it ollows tht E =, which mens E = nd E 2 = 2 nd thereore () nd () re pushouts in nd, respectively. From Fcts nd 2 we otin immeditely the ollowin Fct 3. Fct 3 (omponentwise POs in Product, Slice, oslice teories).since = omt(! :,! :, ), \X = omt(id :, X :, {})

8 372 U. Prne et l. nd X\ = omt(x :, id :, {}) nd the unctors!,id nd X preserve pushouts, it ollows tht lso in product, slice nd coslice cteories pushouts lon M-morphisms re constructed componentwise. 3 dditionl Properties nd Their Preservtion There re severl importnt results in the theory o dhesive HLR systems in [3] where we need (wek) dhesive HLR cteories which stisy some dditionl properties. In this section we nlyze, under which conditions these properties re preserved y the cteoricl constructions discussed in Section 2.Section3. hndles initil pushouts, Section 3.2 hndles inite coproducts comptile with M nd in Section 3.3, the preservtion o n E -M pir ctoriztion is descried. These properties re needed in the theory o dhesive HLR systems s discussed in hpters 5 nd 6 o [3], where coproducts re used or prllel productions, initil pushouts or the extension o trnsormtions nd locl conluence, nd pir ctoriztions or concurrent productions nd criticl pirs. For the existence o initil pushouts, we need specil morphism clss M, since in some cteories initil pushouts over enerl morphisms do not exist or hve more complicted structure tht cnnot e derived rom the cteoricl constructions. This morphism clss M is lso used or the E -M pir ctoriztion, where E is clss o morphism pirs with the sme codomin. For the locl conluence theorem in [3], we need n E -M pir ctoriztion with initil pushouts over M -morphisms. 3. Initil Pushouts n initil pushout ormlizes the construction o the oundry nd context o morphism. For morphism :, we wnt to construct oundry :, oundry oject, nd context oject, ledin to miniml pushout stisyin n initility property. Rouhly spekin, is the luin o nd the context oject lon the oundry oject. einition 3 (Initil Pushout). Given morphism : in (wek) dhesive HLR cteory, morphism : with M is clled the oundry over i there is pushout complement o nd such tht () is pushout which is initil over. Initility o () over mens, tht or every pushout (2) with M there exist unique morphisms : nd c : E with, c M such tht =, c c = c nd (3) is pushout. () (3) (2) c c E c c Remrk I n initil pushouts exists only over M or specil clss M o morphisms, we sy tht (, M) hs initil pushouts over M.

9 onstruction nd properties o dhesive HLR cteories 373 Exmple 4 Initil pushouts exist in Sets, Grphs, Grphs TG nd Grphs TG (see [3]). For Grphs nd Grphs TG, they cn e constructed y Theorem 2. Theorem 2 (Preservtion o Initil Pushouts). Given (wek) dhesive HLR cteories (, M ) nd (, M 2 ) with initil pushouts over M -ndm 2 -morphisms, respectively, or some morphisms clsses M in nd M 2 in. Then we hve the ollowin results.. hs initil pushouts over M M 2 -morphisms. 2. \X hs initil pushouts over M \X-morphisms. 3. For : M X\ with : X the initil pushout exists, i the initil pushout over in cn e extended to vlid squre in X\ or : X M nd the pushout complement o nd in exists. 4. I F preserves pushouts lon M -morphisms nd G(M 2 ) Isos, then omt(f, G; I) hs initil pushouts over M M 2 - morphisms. 5. I hs intersections o M -suojects (see remrk elow) then [X, ] hs initil pushouts over M -nturl trnsormtions. Remrk cteory hs intersections o M -suojects, i it hs the ollowin kind o limits comptile with M : Given c i : i M with i I or some index set I, then the correspondin dirm hs limit (,(c i : i) i I, c : ) in nd we hve tht c i M or ll i I nd c M. i c i c i c j c (i, j I) c j j Proo. Since = omt(! :,! :, ),! preserves pushouts nd! (M 2 ) {id }=Isos this ollows rom item 4. Then the initil pushout (3) o morphism (, 2 ) : (, 2 ) (, 2 ) M M 2 is the product o the initil pushouts () over in nd (2) over 2 in (, 2 ) (, 2 ) (, 2 ) () (2) 2 2 (3) (, 2 ) (, 2 ) c 2 c 2 2 (, 2 ) (c,c 2 ) (, 2 ) 2. Since \X = omt(id :, X :, {}), id preserves pushouts nd X(M 2 ) = X({id }) ={id X } Isos this ollows rom item 4. Then the initil

10 374 U. Prne et l. pushout (2) over : (, ) (, d ) M in \X is iven y the initil pushout () over in,with = nd c = d c. (, ) (, ) () (2) c (, c ) c (, d ) 3. Given ojects (, ), (, d ) nd morphism : in X\ with M. Then the initil pushout () over in exists y ssumption. The remrk o Fct 2 implies tht, or ny pushout (2) in X\ with d, e M, the dirm (3) is pushout in.since() is n initil pushout in there exist unique morphisms : E nd c : F such tht d =, e c = c,, c M nd (4) is pushout in. (, ) d (E, e ) d E () (2) (3) c (, d ) e (F, ) e F E (, ) (, ) (, ) (E, e ) (4) (5) (6) c F (, c ) c (, d ) (, c ) c (F, ) (i) I the dirm (5), correspondin to () in, is vlid extension o () in X\, then the remrk o Fct 2 implies tht it is lredy pushout in X\. It remins to show tht (6) is vlid squre in X\.Withd = = = d e nd d ein monomorphism it ollows tht = e nd thus X\, nd nloously c X\. This mens the squre (6), correspondin to (4), is lso pushout in X\. Thereore (5) is the initil pushout over in X\. (ii) I : X M nd the pushout complement o nd in exists, we cn construct the unique pushout complement (7) in, nd with the remrk o Fct 2 the correspondin dirm (8) is pushout in X\. X (X, id X ) (, ) X X h (7) h (8) (9) h H h (H, h ) h (, d ) c X H It remins to show the initility o (8). For ny pushout (2), e : X E is unique with respect to d e =, ecuse d is monomorphism. Since () is n initil pushout in nd (7) is pushout, there re morphisms X : X nd c X : H such tht X, c X M, X =, h c X = c nd (9) is pushout. With e c = c = h c X = h h X = X = d e X = e e X nd e ein monomorphism (9) implies tht there is unique i : H F with c = i c X nd i h = e.it

11 onstruction nd properties o dhesive HLR cteories 375 urther ollows tht e i = h usin the pushout properties o H. y pushout decomposition, then (0) is pushout in nd usin the remrk o Fct 2 the correspondin squre in X\ is lso pushout. Thereore, (8) is n initil pushout over in X\. X X e d E X e E (9) h (7) (3) h (0) c X H c h i e F H i F 4. Given : omt(f, G; I) with = (, 2 ) M M 2.Thenwe hve initil pushouts () over M in with, c M nd () over 2 M 2 in with 2, c 2 M (, 2, op i ) (, 2 ) (, 2, op i ) () c () c 2 2 () (, 2 ) (, 2 ) (, 2, opi ) (c (, 2, opi,c 2 ) ) Since G(M 2 ) Isos, G( 2 ) nd G(c 2 ) exist. eine ojects = (, 2, opi = G( 2 ) opi F( )) nd = (, 2, opi = G(c ) opi F(c )) in omt(f, G; I). Then we hve G( 2 ) opi = G( 2 ) G( 2 ) opi F( ) = opi F( ), G(c 2 ) opi = G(c 2 ) G(c 2 ) opi F(c ) = opi F(c ), G(c 2 ) G( 2 ) opi = G( 2 ) G( 2 ) opi = G( 2 ) opi F( ) = opi F( ) F( ) = opi F(c ) F( ) = G(c 2 ) opi F( ) nd G(c 2 ) ein n isomorphism implies tht G( 2 ) opi = opi F( ), which mens tht = (, 2 ), = (, 2 ) nd c = (c, c 2 ) re morphisms in omt(f, G; I) with, c M M 2. We shll show tht () is n initil pushout over (, 2 ) in omt(f, G; I). Fct 2 implies tht () is pushout with (, 2 ), (c, c 2 ) M M 2. It remins to show the initility. For ny pushout (2) in omt(f, G; I) with (d, d 2 ), (e, e 2 ) M M 2, Fct 2 implies tht the components (2) nd (2) re pushouts in nd, respectively. (, 2, op i ) (d,d 2 ) (E, E 2, op E i ) d E 2 d 2 E 2 (2) (, 2 ) (, 2 ) (, 2, opi ) (e (F, F 2, opi F,e 2 ) ) (2) e F (2) e 2 F 2 The initility o pushout () implies tht there re unique morphisms : E nd c : F with d =, e c = c nd, c M such tht (3) is pushout. nloously, the initility o pushout () implies tht

12 376 U. Prne et l. there re unique morphisms 2 : 2 E 2 nd c 2 : 2 F 2 with d 2 2 = 2, e 2 c 2 = c 2 nd 2, c 2 M 2 such tht (3) is pushout. (, 2, op i ) (, 2 ) (E, E 2, op E i ) E 2 2 E 2 (3) (, 2 ) (, 2 ) (, 2, opi ) (c (F, F 2, opi F,c 2 ) ) (3) c F (3) c 2 F 2 With G(d 2 ) G( 2 ) op i = G( 2 ) opi =opi F( )=opi F(d ) F( ) = G(d 2 ) opi E F( ) nd G(d 2) ein n isomorphism it ollows tht (, 2 ) omt(f, G; I), nd nloously (c, c 2 ) omt(f, G; I). This mens tht we hve unique morphisms (, 2 ), (c, c 2 ) M M 2 omt(f, G; I) with (d, d 2 ) (, 2 ) = (, 2 ) nd (e, e 2 ) (c, c 2 ) = (c, c 2 ), nd y Fct 2 (3) composed o (3) nd (3) is pushout. Thereore () is the initil pushout over in omt(f, G; I). 5. Let M unct denote the clss o ll M -nturl trnsormtions. Given n M - nturl trnsormtion : in [X, ], y ssumption we cn construct componentwise the initil pushout ( x ) over (x) in or ll x X, with 0 (x), c 0 (x) M. 0 (x) 0 (x) (x) 0 (x) c 0 (x) (x) 0 (x) ( x ) (x) d i (x) (2) c i(x) 0 (x) c 0 (x) (x) i (x) eine (,(c i : i) i I, c : ) s the limit in [X, ] o ll those c i : i M unct such tht or ll x X there exists d i (x) : 0(x) i (X) M with c i (x) d i (x) = c 0(x) (2), which deines the index set I. Limits in [X, ] re constructed componentwise in, ndihs intersections o M -suojects it ollows tht lso [X, ] hs intersections o M unct -suojects. Hence c i M unct nd c M unct,nd(x) is the limit o c i (x) in. Now we construct the pullck (3) over c M unct nd in [X, ] nd since M unct -morphisms re closed under pullcks, lso M unct. 0 (x) (x) 0 (x) (x) (x) (x) (3) 0 (x) (4 x ) (x) (3 x ) (x) c 0 (x) c (x) (x) c(x) (x) d i (x) c i (x) c i(x) i (x) For x X, (x) ein the limit o c i (x), the mily (d i (x)) i I with (2) implies tht there is unique morphism c (x) : 0 (x) (x) with c i (x) c (x) = d i (x) nd c 0 (x)

13 onstruction nd properties o dhesive HLR cteories 377 c(x) c (x) = c 0 (x). Then(3 x ) ein pullck nd c(x) c (x) 0 (x) = c 0 (x) 0 (x) = (x) 0 (x) implies the existence o unique (x) : 0 (x) (x) with (x) (x) = 0 (x) nd (x) (x) = c (x) 0 (x). M is closed under decomposition, 0 (x) M nd (x) M implies (x) M.Since( x ) is pushout, (3 x ) is pullck, the whole dirm commutes nd c(x), (x) M,theM pushout-pullck property (see [3]) implies tht (3 x ) nd (4 x ) re oth pushouts nd pullcks in nd hence (3) nd (4) re oth pushouts nd pullcks in [X, ]. It remins to show the initility o (3) over. Given pushout (5) with, c M unct in [X, ], (5 x ) is pushout in or ll x X. Since( x ) is n initil pushout in, there exist morphisms (x) : 0(x) (x), c : 0(x) (x) with (x), c (x) M, (x) (x) = 0(x) nd c (x) c (x) = c 0(x). Hence c (x) stisies (2) or i = nd d (x) = c (x). 0 (x) 0 (x) (x) (x) (x) (x) (5) 0 (x) ( x ) (x) (5 x ) (x) c 0 (x) c 0 (x) (x) c (x) c (x) (x) This mens c is one o the morphisms the limit ws uilt o nd there is morphism c : with c c = c y construction o the limit. Since (5) is pushout lon M unct -morphisms it is lso pullck, nd = c = c c implies tht there exists unique : with = nd = c unct.ym -decomposition lso M unct. Now usin lso c M unct the M unct pushout-pullck decomposition property implies tht lso (6) is pushout, which shows the initility o (3). (3) (5) (6) (5) c c c c c c Exmple 5 ccordin to the ive cses in Theorem 2 we hve the ollowin exmples sed on Exmple 3, where in ll cses except 3 M is the clss o ll morphisms in.. Initil pushouts in Grphs Grphs cn e constructed componentwise in Grphs (see item 5). 2. Grphs TG hs initil pushouts to e constructed s in Grphs. 3. In PSets ={}\Sets, we hve three dierent cses or initil pushouts over morphism : (, ) (, d ): se : The morphism is injective. Then () is the initil pushout over in Sets with inc ein n inclusion, which cnnot e extended to vlid squre in {}\Sets. ut :{} is injective nd thus M,ndthe pushout complement (2) o nd in Sets exists with = \ ()

14 378 U. Prne et l. ( ()) nd d () = ( ()), which mens y construction in the proo o Theorem 2.3 tht (3) is the initil pushout over in {}\Sets. {} ({}, id) (, ) () d (2) d (3) \ () inc inc (, d ) inc (, d ) se 2: is noninjective, nd () Id( ), withid( ) ={x y, x = y : (x) = (y)}.then(4) with = \ () (Id( )) is the initil pushout over in Sets, which due to d () = ( ()) cn e extended to vlid dirm (5) nd hence y construction to the initil pushout (5) over in {}\Sets. Id( ) inc (Id( ), ) inc (, ) Id( ) (4) Id( ) (5) inc (, d ) inc (, d ) se 3: is noninjective, nd () / Id( ). In this cse, pushout (4) ove is the initil pushout over in Sets, ut it cnnot e extended to vlid squre in {}\Sets. Moreover, the pushout complement over nd does not exist in Sets, thus Theorem 2 cnnot e pplied. Nevertheless, the initil pushout in this cse in {}\Sets exists, it is the ollowin pushout (6) with = \ () (Id( )) d (), which enerlises cses nd 2. (Id( ) (), ) Id( ) () inc (6) (, ) (, d ) inc (, d ) 4. The constructions in product nd slice cteories re exmples or the construction o initil pushouts in comm cteories; hence we cn reuse the exmples in items nd Since Sets hs intersections o M-suojects or the clss M o injective unctions, the cteory Grphs s unctor cteory o Sets hs initil pushouts. 3.2 Finite oproducts omptile with M In the doule pushout rph trnsormtion, or prllel productions we need not only inite coproducts or the deinition o the ojects o the prllel production, ut lso inite coproducts comptile with M which ensure tht the morphisms o the prllel productions re M-morphisms. For the existence o inite coproducts comptile with M it is suicient to show the existence o inry coproducts comptile with M.

15 onstruction nd properties o dhesive HLR cteories 379 einition 4 (inry oproduct omptile with M). (wek) dhesive HLR cteory (, M) hs inry coproducts comptile with M i hs inry coproducts nd, or ech pir o morphisms :, : with, M, the coproduct morphism is lso n M-morphism, i.e. + : + + M. i + i + i + i Exmple 6 Sets, Grphs, Grphs TG nd Grphs TG hve inry coproducts comptile with M. For Grphs nd Grphs TG this ollows y Thm. 3, or Grphs TG see [3]. Theorem 3 (Preservtion o in. oproducts omptile with M). Given(wek) dhesive HLR cteories (, M ) nd (, M 2 ) with inry coproducts comptile with M nd M 2, respectively. Then we hve the ollowin results.. hs inry coproducts comptile with M M \X hs inry coproducts comptile with M \X. 3. X\ hs inry coproducts comptile with X\ M, i hs enerl pushouts. 4. I F preserves coproducts, then omt(f, G; I) hs inry coproducts comptile with M M 2 -morphisms. 5. [X, ] hs inry coproducts comptile with M -nturl trnsormtions (see remrk ter Theorem 2). Proo. Since = omt(! :,! :, ) nd! preserves coproducts this ollows rom item 4. The coproduct o ojects (, 2 ) nd (, 2 ) o the product cteory is the componentwise coproduct ( +, ) in nd, respectively. nloously, the coproduct o morphism = (, 2 ) nd = (, 2 ) is the componentwise coproduct morphism + = ( +, ) in nd. I, M M 2,thenlso + M M Since \X = omt(id :, X :, {}) nd id preserves coproducts this ollows rom item 4. In the slice cteory, the coproduct o (, ) nd (, ) is the oject ( +, [, ]) which consists o the coproduct + in toether with the morphism [, ]: + X induced y nd. nloously, or morphisms nd the coproduct morphism in \X is the coproduct morphism + in. I, M \X, thenlso + M \X. 3. I hs enerl pushouts, iven two ojects (, ) nd (, ) in X\ we construct the pushout () over nd in. The coproduct o (, ) nd (, ) is the pushout oject + X. Given morphisms : nd : in X\, we cn construct the pushouts (), (2), (3) nd (4) in, nd y pushout composition we hve G = + X ledin to the coproduct morphism + X

16 380 U. Prne et l. in X\. Usin the utterly Lemm (see [3, 0]), lso (5) is pushout in. I, M, + M ecuse hs inry coproducts comptile with M. Since M -morphisms re closed under pushouts, it ollows tht lso + X M. d X () (2) c + X E (5) (3) (4) + X + X + X + X F G = + X 4. I nd hve inry coproducts nd F preserves coproducts, then the coproduct o two ojects = (, 2, opi ) nd = (, 2, opi ) in omt(f, G; I) is the oject + = ( +, 2 + 2, op + i ), where op + i is the unique morphism induced y G(i 2 ) opi nd G(i 2 ) opi.ig preserves coproduct, i.e. G( 2 )+G( 2 )= G( ),thenop + i =opi +opi. F( ) F(i ) F( + ) F(i ) F( ) op i op + i op i G( 2 ) G(i 2 ) G( ) G(i 2 ) G( 2 ) For morphisms = (, 2 ) : (, 2, opi ) (, 2, opi ) nd = (, 2 ) : (, 2, opi ) (, 2, opi ) we et coproduct morphism + = ( +, ).I, M M 2, the comptiility o coproducts with M nd M 2 in nd ensures tht + M nd M 2, respectively, tht mens + M M I hs inry coproduct, the coproduct o two unctors, : X in [X, ] is the componentwise coproduct unctor + with + (x) = (x) + (x) or n oject x X nd + (h) = (h) + (h) or morphism h X. For nturl trnsormtions = ( x ) x X nd = ( x ) x X, the coproduct morphism is the componentwise coproduct morphism in,i.e. + = ( x + x ) x X.I nd re M -nturl trnsormtions we hve x, x M or ll x, nd since hs inry coproducts comptile with M it ollows tht lso x + x M or ll x X, thereore + is n M -nturl trnsormtion. Exmple 7. In Grphs Grphs, inry coproducts re constructed componentwise nd re comptile with injective morphisms. 2. In Grphs TG, inry coproducts re constructed in Grphs nd lited to Grphs TG. From the coproducts in Grphs they inherit the comptiility with injective morphisms.

17 onstruction nd properties o dhesive HLR cteories The construction o inry coproducts in PSets ={}\Sets is iven y the correspondin pushout in Sets, i.e. the inry coproduct o two sets nd with distinuished elements nd, representin the respective undeined, is the set \{ } \{ } {},where represents the new undeined element o the coproduct. 4. In PTNets, inry coproducts re constructed componentwise in Sets nd re comptile with injective Petri net morphisms. 5. In Grphs, inry coproducts re constructed componentwise in Sets nd re comptile with injective rph morphisms. 3.3 E -M Pir Fctoriztion possile pproch to nlyze the conluence o trnsormtion system is to show the termintion o the system, nd the strict conluence o so-clled criticl pirs. The concept o n E M pir ctoriztion is essentil or the deinition o criticl pirs. einition 5 (E M Pir Fctoriztion). Given clss o morphism pirs E with the sme codomin nd clss M o morphisms, (wek) dhesive HLR cteory hs n E M pir ctoriztion i, or ech pir o morphisms : nd :, there exist n oject K nd morphisms e : K, e : K,ndm : K with (e, e ) E nd m M such tht m e = nd m e = : e e K m einition 6 (Stron E M Pir Fctoriztion). n E M pir ctoriztion is clled stron, i the ollowin E M dionl property holds: Given (e, e ) E, m M, nd morphisms,, n s shown in the ollowin dirm, with n e = m nd n e = m, then there exists unique d : K L such tht m d = n, d e = nd d e =. e e d K n L m Fct 4 (Stron E M Pir Fctoriztion). In (wek) dhesive HLR cteory (, M), the ollowin properties hold:. I (, M) hs stron E M pir ctoriztion, then the E M pir ctoriztion is unique up to isomorphism. 2. stron E M pir ctoriztion is unctoril, i.e. iven morphisms,, c,,, 2, 2 s shown in the ollowin dirm with c = 2 nd

18 382 U. Prne et l. c = 2, nd E M pir ctoriztions ((e, e ), m ) nd ((e 2, e 2 ), m 2) o, nd 2, 2, respectively, then there exists unique d : K K 2 such tht d e = e 2, d e = e 2 ndc m = m 2 d. e e K m 2 2 d c e 2 K 2 m 2 2 e Proo. We show tht stron E M pir ctoriztions re unique up to isomorphism. Suppose ((e, e ), m ) with m : K nd ((e 2, e 2 ), m 2) with m 2 : K 2 re two E M pir ctoriztions o nd. e e e 2 e 2 K k k K 2 m m 2 Usin the morphisms (e, e ) E nd m M,romtheE M dionl property we otin unique morphism k : K K 2 with m 2 k = m, k e = e 2 nd k e = e 2. t the sme time, id K is unique with respect to m id K = m, id K e = e nd id K e = e. y exchnin the roles o ((e, e ), m ) nd ((e 2, e 2 ), m 2), thee M dionl property implies tht there is unique k : K 2 K with m k = m 2, k e 2 = e nd k e 2 =e.lso,id K 2 is unique with respect to m 2 id K2 = m 2, id K2 e 2 = e 2 nd id K2 e 2 = e 2. Thus, rom m k k = m 2 k = m, k k e = k e 2 = e nd k k e = k e 2 = e it ollows tht k k = id K, nd nloously k k = id K2. This mens tht K nd K 2 s well s the correspondin morphisms re isomorphic. 2. The ct tht stron E M pir ctoriztion is unctoril ollows directly rom the E M dionl property: Given the settin ove, since (e, e 2 ) E

19 onstruction nd properties o dhesive HLR cteories 383 nd m 2 M we otin unique morphism d : K K 2 with m 2 d = c m, d e = e 2 nd d e = e 2. Exmple 8 In Sets, one possile choice or E nd M is to deine E s the clss o jointly surjective morphisms nd M s the clss o injective morphisms ledin to stron E M pir ctoriztion. In cteories with inry coproducts nd n E M ctoriztion we hve n E M pir ctoriztion with M = M nd (e, e ) E [e, e ] E,where[e, e ] is the morphism induced y the inry coproduct. In the cteory Grphs TG there re dierent possile choices or n E M pir ctoriztion (see [3]). Theorem 4 (Preservtion o E M Pir Fctoriztions). Given (wek) dhesive HLR cteories (, M ) nd (, M 2 ) with E M nd E 2 M 2 pir ctoriztions, respectively. Then we hve the ollowin results.. hs n E M pir ctoriztion with M = M M 2 nd E ={((e, e 2 ), (e, e 2 )) (e, e ) E,(e 2, e 2 ) E 2 }.ItheE M nd E 2 M 2 pir ctoriztions re stron, so is the E M pir ctoriztion. 2. \X hs n E M pir ctoriztion, where stronness is preserved. 3. I M is clss o monomorphisms then X\ hs n E M pir ctoriztion, where stronness is preserved. 4. I G(M 2 ) Isos, then omt(f, G; I) hs n E M pir ctoriztion (with E, M s in the product cteory), where stronness is preserved. 5. I E M unct is stron pir ctoriztion in, then[x, ] hs stron E M unct pir ctoriztion, where M unct is the clss o M -nturl trnsormtions nd (e, e ) E unct i (e(x), e (x)) E or ll x X. Proo. Since = omt(! :,! :, ) nd! (M 2 ) {id }=Isos this ollows rom item 4. For morphisms = (, 2 ) nd = (, 2 ) in we construct the componentwise pir ctoriztions ((e, e ), m ) o, with (e, e ) E nd m M nd ((e 2, e 2 ), m 2) o 2, 2 with (e 2, e 2 ) E 2 nd m 2 M 2. This leds to morphisms e = (e, e 2 ), e = (e, e 2 ) nd m = (m, m 2 ) in nd n E M pir ctoriztion with (e, e ) E nd m M. ItheE M nd the E 2 M 2 pir ctoriztions re stron, then lso E M is stron pir ctoriztion. 2. Since \X = omt(id :, X :, {}) nd X(M 2 ) X({id }) = {id X } Isos this ollows rom item 4. Given morphisms nd in \X, n E M pir ctoriztion o nd in is lso n E M o nd in \X. I the E M pir ctoriztion is stron in, this is lso true or \X. 3. Given morphisms : (, ) (, c ) nd : (, ) (, c ) in X\, we hve n E M pir ctoriztion ((e, e ), m) o nd in. Thisislsopir ctoriztion in X\ i e = e, ecuse then (K, e ) nd (K, e ) is

20 384 U. Prne et l. the sme oject in X\. Im is monomorphism, this ollows rom m e = = c = = m e. X e e K m To show tht stronness is preserved, we hve to show the E M dionl property in X\. Since it holds in, iven (e, e ) E X\, m M X\ nd morphisms,, n in X\ with n e = m nd n e = m we et n induced unique d : K L with d e =, d e = nd m d = n. It remins to show tht d is vlid morphism in X\. (, ) e (, ) e (K, k ) d (L, l ) m (, c ) Since m d k = n k = c = m l nd m is monomorphisms it ollows tht d k = l nd thus d X\. 4. Given ojects = (, 2, opi ), = (, 2, opi ), = (, 2, opi ) nd morphisms = (, 2 ) :, = (, 2 ) : in omt(f, G; I), we hve n E -M pir ctoriztion ((e, e ), m ) o, with m : K in nd n E 2 -M 2 pir ctoriztion ((e 2, e 2 ), m 2) o 2, 2 with m 2 : K 2 2 in. I G(m 2 ) is n isomorphism, we hve n oject K = (K, K 2, opi K = G(m 2 ) opi F(m )) in omt(f, G; I). y deinition, m = (m, m 2 ) : K is morphism in omt(f, G; I). For e = (e, e 2 ) we hve opi K F(e ) = G(m 2 ) opi x F(m ) F(e ) = G(m 2 ) opi F( ) = G(m 2 ) G( 2 ) opi = G(e 2 ) opi nd n nloous result or e = (e, e 2 ), thereore e nd e re morphisms in omt(f, G; I). This mens, ((e, e ), m) is n E -M pir ctoriztion in omt(f, G; I). n F( ) F( ) F(e ) F(K ) F(m ) F( ) op i F(e ) F( ) F( ) G( 2 ) opi K =G(m opi 2 ) opi F ( m ) G( 2 ) op i G(e 2 ) G(K 2 ) G(m 2 ) G( 2 ) G(e 2 ) G( 2 ) G( 2 )

21 onstruction nd properties o dhesive HLR cteories 385 To show the E M dionl property, we consider (e, e ) = ((e, e 2 ), (e, e 2 )) E, m = (m, m 2 ) M nd morphisms = (, 2 ), = (, 2 ), n = (n, n 2 ) in omt(f, G; I). (, 2, op i ) (, 2 ) (e,e 2 ) (, 2, opi ) (e (K, K 2, opi K,e 2 ) ) (, 2 ) (d,d 2 ) (n,n 2 ) (L, L 2, opi L ) (m (, 2, opi,m 2 ) ) Since (e, e ) E nd m M, we et uniqe morphism d : K L in with m d = n, d e = nd d e =. nloously, the E 2 M 2 dionl property implies unique d 2 : K 2 L 2 with m 2 d 2 = n 2, d 2 e 2 = 2 nd d 2 e 2 = 2. It remins to show tht d = (d, d 2 ) omt(f, G; I), i.e. the comptiility with the opertions. We hve or ll i I G(m 2 ) opi L F(d ) = opi F(m ) F(d ) = opi F(n ) = G(n 2 ) opi K = G(m 2 ) G(d 2 ) opi K, nd since G(m 2 ) is n isomorphism it ollows tht opi L F(d ) = G(d 2 ) opi K,i.e.d omt(f, G; I). F(K ) G(K 2 ) op K i F(d ) op L i G(d 2 ) F(L ) F(n ) G(n 2 ) G(L 2 ) F(m ) G(m 2 ) F( ) G( 2 ) op i 5. Given morphisms = ( (x)) x X nd = ((x)) x X in [X, ], we hve n E M pir ctoriztion ((e x, e x ), m x) with m x : K x (x) o (x), (x) in or ll x X. We hve to show tht K(x) = K x cn e extended to unctor nd tht e = (e x ) x X, e = (e x ) x X nd m = (m x ) x X re nturl trnsormtions. For morphism h : x y in X, weusethee M dionl property in nd (e x, e x ) E, m y M to deine K h : K x K y s the unique induced morphism with m y K h = (h) m x, K h e x = e y (h) nd K h e x = e y (h).

22 386 U. Prne et l. (x) (x) e x (h) e x K x m x (x) (x) (x) (y) (y) K h (h) (h) e y K y m y (y) (y) e y (y) Usin the uniqueness property o the stron pir ctoriztion in, we cn show tht K with K(x) = K x, K(h) = K h is unctor nd y construction e, e nd m re nturl trnsormtions. This mens (e, e ) E unct nd m M unct,i.e.this is n E unct M unct pir ctoriztion o nd. The E unct M unct dionl property cn e shown s ollows. Given (e, e ) E unct, m M unct nd morphisms,, n in [X, ], romthee M dionl property in we otin unique morphism d x : K(x) L(x) or x X. It remins to show tht d = (d x ) x X is nturl trnsormtion, i.e. we hve to show or ll h : x y X tht L(h) d x = d y K(h). (h) (y) (x) (x) (x) (x) e(x) e (x) d x K(x) n(x) (h) e(y) K(h) (y) (y) (y) e (y) d y K(y) n(y) L(x) m(x) (x) L(y) m(y) (y) L(h) (h) onsider the ollowin dirm, where ecuse o (e(x), e (x)) E nd m(y) M the E M dionl property cn e pplied. This mens there is unique k : K(x) L(y) with k e(x) = L(h) (x), k e (x) = L(h) (x) nd m(y) k = n(y) K(h). (x) (x) L(y) e(x) L(h) (x) L(h) (x) e (x) k m(y) K(x) n(y) K(h) (y)

23 onstruction nd properties o dhesive HLR cteories 387 For L(h) d x we hve: L(h) d x e(x) = L(h) (x), L(h) d x e (x) = L(h) (x) nd m(y) L(h) d x = (h) m(x) d x = (h) n(x) = n(y) K(h). For d y K(h) we hve: d y K(h) e(x) = d y e(y) (h) = (y) (h) = L(h) (x), d y K(h) e (x) = d y e (y) (h) = (y) (h) = L(h) (x) nd m(y) d y K(h) = n(y) K(h). Thus, rom the uniqueness o k it ollows tht k = L(h) d y = d y K(h) nd d is nturl trnsormtion. Exmple 9. In Grphs Grphs, the stron E M pir ctoriztion is constructed componentwise in Grphs (see item 5). 2. The cteory Grphs TG inherits the stron E M pir ctoriztion rom Grphs. 3. The cteory PSets ={}\Sets inherits the stron E M pir ctoriztion rom Sets,whereM is the clss o ll injective morphisms nd E is the clss o jointly surjective morphisms. 4. The constructions in product nd slice cteories re exmples or the construction o n E M pir ctoriztion in comm cteories; hence we cn reuse the exmples in items nd The cteory Grphs inherits the stron E M pir ctoriztion rom Sets in Exmple 8. It is constructed componentwise on the node nd ede sets. 4 onclusion nd Future Work The leric theory o rph trnsormtions [5, 7] hs een enerlized recently to the rmework o dhesive hih-level replcement systems, which re sed on dhesive [, 2] nd dhesive hih-level replcement cteories [3, 6]. It hs een shown lredy tht this kind o cteories is closed under product, slice, coslice, comm nd unctor cteory constructions. The min contriution o this pper is to show under which conditions dditionl properties like initil pushouts, coproducts nd pir ctoriztions, needed in the theory o dhesive hih-level replcement systems, re preserved under these constructions. This voids to check these properties explicitly or ll the instntitions o dhesive hih-level replcement cteories. The most importnt new results re the construction o initil pushouts in unctor nd comm cteories, ecuse in enerl initil pushouts cnnot e constructed componentwise. Moreover pir ctoriztions, introduced in [3, 6] s modiictions o E M ctoriztions, re in enerl not preserved y unctor cteory constructions. For this reson we hve introduced the new notion o stron pir ctoriztion, which requires in ddition dionl property similr to tht o E M ctoriztions. This llows to show tht stron pir ctoriztions re preserved y ll the cteory constructions includin unctor cteories. oncernin uture work it remins to relx some o the conditions under which the dditionl properties re preserved. Moreover, it is interestin to nlyze how to construct pushout complements in dhesive hih-level replcement cteories, where chrcteriztion or the existence o pushout complements usin initil pushouts is iven lredy in [3].

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