.P.. ci. ull. eries ol. 70 No. 3 2008 IN 223-7027 IN ELTION DDEND KENEL ETICTION ND INDCING ELTIONL MOPHIM Mihai EENCIC ceastă lucrare în două părţi conţine unele completări la teoria relaţiilor binare într-un cadru extins prin operaţii categoriale generalizate relativ la categoria neregulată a relaţiilor binare el.asociată categoriei regulate et. Primele două completări din această parte a lucrării se referă la nucleul respectiv la restricţiile şi indusa în mulţimi arbitrare a unei relaţii binare în legătură cu operaţii de algebră ooleană şi categoriale generalizate ltima completare constă într-o ierarhie de morfisme relaţionale în paralel în cazurile omogen şi neomogen la care se raportează şi noţiunea de bisimulare generalizată pentru cazul neomogen esenţială în programarea concurentă. This paper in two parts contains some addenda to binary relations theory in a background witch is extended by generalized categorical operations relative to the unregulated category of binary relations el associated with the regulate category et. The first two addenda from this part of the paper refer to the kernel respectively to the restrictions and the induced relation in arbitrary sets of a binary relation - in connection with oolean algebra operations and generalized categorical operations. The last addendum consists in a hierarchy of relational morphisms in parallel in homogeneous and inhomogeneous cases to witch the notion of bisimulation generalized for inhomogeneous case is reported - important in concurrency programming. Keywords: category of relations relational systems category of sets. MC2000: primary80 0802; secondary 805.. Introduction The concept of multivocity is illustrated in a naturally way by the notion of binary relation in the regulate category of sets et in witch the binary relations form the unregulated category el in regard to the categorical operation of composition and the binary relations between the same sets have a structure of complete oolean algebra with the known properties [] [2] [3] [4]; in a topoi E it is maintained the corresponding category el E but the structure of a Lecturer Department of Mathematics III niversity Politehnica of ucharest OMNI e-mail: m_rebenciuc@yahoo.com
2 Mihai ebenciuc complete oolean algebra it is replaced by the structure of Heyting algebra [5] and in a category that in the finite case is more generally then a topoi wellpowered and wellcopowered category with products coproducts and finite intersections remains just the structure of complete lattice - with a conditionally distributivity of the composition in regard to union [6] [7]. The first addendum from this part of the paper is relates to the functional type by defining the kernel of a relation and the determination of some properties as a special homogeneous relation of tolerance and equivalence respectively of connection with generalized categorical operations. In the second addendum we associate the concept of multivocity with the one of partiality - with a unification of functional and order approach by defining in the same generalized way of the notions of restrictions and induced binary relation in arbitrary sets followed by the study of oolean algebra and generalized categorical operations. These generalisations - relative to categorical operations and restrictions and inducing in arbitrary sets are categorical validated in categories with intersections and unions of objects [8] [9]. The last addendum refers to a hierarchy of relational morphisms with a parallel between homogeneous and inhomogeneous cases - to witch the notion of bisimulation generalized for inhomogeneous case is reported ; the notion of bisimulation that induces the notion of bisimilarity is important in concurrency programming [0] []. We close up with an example relative to ones of the above generalizations. Example. operations with subtotal and subdiagonal relations elative to the set el of binary relations between we consider the set el st { / P P} el of the subtotal relations with the total relation and for P * P * with the vide relation in el ; τ : P{ } el st τ and analogously τ 2 is complete isomorphism of oolean algebras but τ: P P el st τ remains only morphism of inferior semilattices and partial order semiembedding with P P complete oolean algebra and with el st bounded inferior semilattice by Ø. Consequently el st el st are complete oolean subalgebras of complete oolean algebra el - through the medium of
inary relations addenda Kernel restrictions and inducing relational morphisms 3 inferior semilattice el st ; analogously in homogeneous case el el - in addition with el sd { {x x/x }/ P} el the set of subdiagonal relation in with the diagonal relation in complete sublattice of el according to complete lattice isomorphism δ: P el sd δ. el tot corresponding to the sets el tot { } el tot { } is preordered subcategory of el with ; categorical operation of composition graphically omitted is generalized by for C D C respectively C D D for C. If el is nonstrictly i.e. O [2] then el is semigroup relative to composition the neutral elements to the right respectively to the left which el st subsemigroup. In homogeneous case el is monoid with el sd submonoid because ; in addition we have. 2. Kernel Definition 2. cokernel of a relation. The kernel of the relation el is the homogeneous relation of el noted ker and defined by ker ; dually co ker el is the cokernel of. Observation 2..i duality We have co ker ker ker co ker because co ker ker and analogously for the other equality. ii symmetry Ker is symmetric relation because ker ker and analogously for cokernel. iiitolerance Ker is D-tolerance relation D dom subfieldker domker codomker where domker codomker dom because ker is reflexive in any a D - there is b such that and hence ker and according to ii; analogouslybut dually coker is C-tolerance relation C codom subfieldcoker. Particularly if is left-total dom respectively right-total codom then ker respectively coker are tolerance relations.
4 Mihai ebenciuc iv equivalence If in addition el is dysfunctional [3] i. e. is a sufficient condition which is named also i-regulate or preunivocalfor unification of terminology in the category el [4] then ker is D-equivalence and coker is C-equivalence; in the case left-total respectively right-total these became equivalences. Indeed we have 2 ker ker and analogously for cokernel. Theorem 2.inclusion categorical operations Let be el ; for el implies coker coker. For elc D we have the equalities ker ker ker coker coker coker coker coker. Proof. elative to inclusion we have ker ker and analogously for cokernel. elative to categorical operations we have ker ker ker and analogously for cokernel. In addition we have ker iff there exists b d D such that a b d b d iff d d iff a ker ker respectively b d d co ker iff there exists a C such that b d d iff a d d iff b co ker d d coker iff b d d co ker co ker. Definition 2.2w-composability The relations el elc D are weak composable for short w-composable if there exist composable pairs of i. e... Particularly is weak self-composable for short w-selfcomposable if 2 * ; more generally for n IN \ { implicitly n 2 n is n-weak self-composable for short nw-self-composable if. Observation 2.2.i sufficient condition relation is w-self-composable if it is non-banal transitive. * ii monotony For m n IN \ { if m < n then n-w-self-composable implies m-w-self-composable. iii the w-self-composability of the cokernel Ker and analogously coker is w-self-composable because whichever of the inclusion or
inary relations addenda Kernel restrictions and inducing relational morphisms 5 D ker D dom see the points iii and iv of observation 2. imply the 2 inclusion ker ker with ker. 3. estrictions and inducing Definition 3. corestriction induced relation Let el be a relation and let be arbitrary sets; the restriction of to and dually the corestriction of to and the relation which is induced by in are respectively the relations el. Observation 3. nuances terminology More exactly we have el el which are named also the left restriction of to respectively the right restriction of to and el ; is the extension of and respectively to and. For el is the relation which is induced by in. Theorem 3. connections Let el be a relation and let be arbitrary sets. We have the equalities - associativity. Proof. The first two equalities the last equality and the equalities of the associativity follow at once by definition or according to the last equality for the equalities of the associativity and imply the equality. Finally we have. Example 3. restrictions and induced relations of the subtotal and subdiagonal relations elative to the arbitrary sets we have the following restrictions and induced relations of the relations el st el sd : - particularly for P * P * respectively - particularly for P * \ \. In addition for el we have
Mihai ebenciuc 6 Theorem 3.2 relations and operation of oolean algebr Let el be relations and let ' ' be arbitrary sets. i inclusion preserving imply imply - and analogously it is the inclusion preserving relative to corestriction imply imply ; ii union preserving - and analogously it is the union preserving relative to corestriction ; iii behaviour towards intersection - and analogously for corestriction. Proof. i. It is easy for example by making use of the expressions of the restrictions and of the induced relation with subdiagonal relations as operands of the composition see theorem 3. and by inclusion preserving by composition. ii. We have see theorem 3. or by definition and according to the example. - and analogously for the preservation of the union relative to corestriction. iii. We have by definition - and analogously for the other expression and for corestriction - and analogously for the other expressions.
inary relations addenda Kernel restrictions and inducing relational morphisms 7 Observation 3.2 the taking of the restrictions and of the induced relation as morphisms The taking of the restriction ρ :P el el χ : el P el respectively of the induced relation ι : P el P el are order morphisms and lattice morphisms in the relation argument; they only are the superior semilattice morphisms in the set argument respectively in an set argument. Theorem 3.3 generalized categorical operations Let el elc D be relations and let be arbitrary sets. i inversion - and analogously for corestriction ; ii composition - and analogously for corestriction ; iii other operations - and analogously for corestriction. Proof. i. We have see theorem 3. - and analogously for corestriction. ii. We have successively see theorem 3. - and analogously for corestriction. iii. We have successively see theorem 3. example. and [4] respectively - and analogously for corestriction. 4. elational morphisms
8 Mihai ebenciuc Definition 4. r-bimorphism The inhomogeneous relation F el is relational morphism for short r-morphism between the homogeneous relational structures if it is compatible with i. e. for each a b with F or equivalently with 2 a b F F F implies b ; F is relational bimorphismrelational semiembedding for short r-bimorphismr-semiembedding if F F are r-morphisms. Observation 4. partial but non-banal compatibility The compatibility with from the r-morphism condition can be partially but it is totally nonbanal because a dom F; other two distinct conditions with F respectively a F after implication lead to two partially non-banal variants. Definition 4.2 variants The inhomogeneous relation F el is r'-morphism between the homogeneous relational structures if for each a b with F implies for each F b ; F is r " -morphism if for each a imply for each b F b. F is r'-bimorphism r'-semiembedding if F F are r'-morphisms and analogously for r " -bimorphism r " -semiembedding. Theorem 4. Let be F el. i connections The condition of r-bimorphism is equivalent with the condition of r-morphism with equivalence instead of implication. The condition of F r'-morphism with equivalence imply F r'-morphism between ; the condition of F r " -morphism with equivalence imply F r-morphism. ii hierarchy We have the implications F r-morphism imply F r'-morphism imply F r " -morphism with equivalences if domf field ; analogously for the relational bimorphisms with the equivalences condition in hierarchy completed with codomf field. Proof. i. The statements follow at once by definition. ii. We have r r r where F r-morphism iff a r a F r'-morphism iff a r F r " -morphism iff a r r t t p a q r t
inary relations addenda Kernel restrictions and inducing relational morphisms 9 p a t q t p a t p a q r p t t q p a t p a t p a q p q b b t b F t F. The conditioned equivalences by dom F field are consequences of the above implications and of the total non-banality of the implication from the condition of r " -morphism because for q b b r a b codomf p q b r p a b codomf q b we have r r. In the case of the relational bimorphisms the above results are valid for the inverse relation F e l with codomf dom F field. Observation 4.2.i categorical composition The categorical composite of two w-composable r-morphisms is r-morphism; analogously for the other relational morphisms and for the relational bimorphisms w-composed under the conditions of the equivalences in relational bimorphisms hierarchy. ii the case of the left-total and right-total relations The left-total relations satisfy the equivalences condition in relational morphisms hierarchy.; the left-total and right-total relations satisfy the equivalences condition in relational bimorphisms hierarchy. iii duality The condition b instead of b lead to dual r-bimorphism with invariant respectively partial variant composite towards dualizing. iv the inhomogeneous case vs. the homogeneous case In the inhomogeneous case relative to the inhomogeneous relational structures a inhomogeneous r-morphism is a ordered pair F F el el' ' with F F F F el which satisfies the compatibility condition with i. e. for each a b with F a F or equivalently with a b F F a implies b ; in fact in homogeneous case a r-morphism is a singlet {F} F F noted F which satisfies the compatibility condition. The other relational morphisms and the relational bimorphisms can be defined similarly. In addition in the inhomogeneous case are valid theorem 4. and points i-iii where F F F F G G F F GF G F codom F F' codomf codomf'.
20 Mihai ebenciuc v bisimulation simulation el between the homogeneous relational structures is defined by a non-banal existential variant of the compatibility condition with of a r'-morphism i. e. for each a b with implies there exists b - which is equivalently with the usual conditionsee [0] [] for each a b with and for each implies there exists b ; but it is weaker than the r-morphism condition which is conditioned equivalently with the r'-morphism conditionsee theorem 4. and point i with. There is analogously for bisimulation vs. r'-bimorphism respectively r-bimorphism with the mention of the non-equivalence between the bisimulation condition and the simulation condition with equivalence instead of implication. Definition 4.3 bisimulation in the inhomogeneous case simulation between the inhomogeneous relational structures is a ordered pair el el' ' with el which satisfies the non-banal existential variant of the compatibility condition with i. e. for each a b with implies there exists a a b ; is bisimulation if it is simulation along with. Observation 4.3.i equivalence The compatibility condition is equivalently with the condition for each a b with and for each implies there exists b a a b a inhomogeneous analogue of the usual condition in homogeneous case see observation 4.2.v. ii two-way similarity In addition can be defined similarity bisimilarity two-way similarity respectively < el by a < b iff there exists el el' ' simulation with a b iff there exists el el' ' bisimulation with a b iff a < b b< a i. e. there exists el el' ' T T el el' ' simulations with b T - with the inclusion. iii strict inclusion Generally the inclusion is strictly see next example hence we have not equality possibly we have only conditioned equality.
inary relations addenda Kernel restrictions and inducing relational morphisms 2 Example 4. counterexample For { a} { a } { a } {b} {} and the simulations el el' ' T T el el' ' T T we have a b and non a b. 5. Conclusions In the first two addenda of this part of the paper we define the notions of kernel and restriction which are dualized respectively the notion of induced relation where the last two are in arbitrary sets - in connection with oolean algebra operations and generalized categorical operations see the theorems 2. 3. 3.2 3.3 and the observation 3.2 ; so that we have done a unification of functional and order approach and more generally an association of the multivocity and partiality concept existent in some domains of the theoretical computer science. These generalizations relative to categorical operations restrictions and inducing in sets are categorical valid [8] [9]. The last addendum refers to a hierarchy of relational morphisms in parallel in homogeneous and inhomogeneous cases - with equivalence conditions see the theorem 4. and the observation 4.2.iv; the notion of bisimulation generalized for the inhomogeneous case is reported to this hierarchy and it is important in concurrency programming [0] [] along with the notion of bisimilarity which it induces. E F E E N C E []. N. Moschovakis et Theory Notes. ndergraduate Texts in Mathematics Heidelberg: pringer erlag 994. [2] M. eghiş Elemente de teoria mulţimilor şi de logicã matematicã. Ed. Facla Timişoara 98. [3] I. Purdea Gh. Pic Tratat de algebrã modernã vol. I Ed. cademiei omâniei ucureşti 977. [4] M. ebenciuc elaţii în categorii logici neclasiceteză de doctorat niversitatea din ucureşti Facultatea de Matematică 998 999. [5]. Gh. adu Teoria toposurilor vol. I Ed. cademiei omâniei ucureşti 98. [6] M. ebenciuc On binary non-homogeneous relations in a category. ull.. P.. eria Metalurgie L -20 988. [7] M. ebenciuc On binary non-homogeneous relations in a category 2. ull.. P.. eria Metalurgie LI -20 989. [8] M. ebenciuc Categories with intersections and unions of objects and with preorder axiomspreprint 2007.
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