G. Birkho. On the structure of abstract algebras. In Proceedings of The Cambridge Philosophical Society, volume 31, pages 433-454, 1935.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....space and therefore so is every closed subspace. Now consider the free monoid A # on the set A and the unique homomorphism # : A # M which maps each letter a to the element whose M component is the corresponding generator of M (M V 0 ) It is a well known result of Birkho# [12] that the image of # is the V free monoid on the set A, which we AV. The closure AV in M is a compact monoid which we AV. It is the completion AV with respect to the ultrametric of Proposition 2.1 and it may also be seen directly as the completion of A # with respect to the ....

G. Birkho#, On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 31 (1935) 433--454.

....that is closed under ultraproducts [12] An equation # (or, more generally, a first order sentence) such that for any algebra A of its signature, A = # implies A = #, is said to be canonical. Any set of equations defines a variety, and conversely, any variety is definable by equations [2]. Clearly, any set of canonical equations defines a canonical variety; but the converse of this is not immediate. Let V be a canonical variety of BAOs, and let # be a set of equations that axiomatises V. Then A = # implies A = #. But it is not apparent that A = # implies A = # for each ....

G. Birkho#, On the structure of abstract algebras, Proc. Cambr. Philos. Soc. 31 (1935), 433--454.

....of coalgebras that is closed under disjoint unions (coproducts) images of coalgebraic morphisms, and subcoalgebras, dualizing the classical notion of a variety as a class of universal algebras closed under direct products, subalgebras, and homomorphic images. A celebrated result of Garret Birkho [7] states that such varieties are precisely the equationally de nable classes of algebras. There has been a spate of papers discussing coalgebraic versions of Birkho s theorem [29, 3, 17, 20, 23, 2, 15] This paper is concerned with covarieties that are closed under images of bisimulations, aptly ....

....so F is natually isomorphic to G . To summarize these correspondences: the assignments K 7 G and F 7 K establish a bijection, up to natural isomorphism, between behavioural covarieties of T coalgebras and tight subcomonads of . 6 Birkho s Theorem Revisited The famous result of [7] shows that a certain logically speci ed notion an equationally de nable class of algebras has a structural characterisation: closure under homomorphic images (H) subalgebras (S) and direct products (P) Classical model theory subsequently developed many such preservation theorems , ....

Garret Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

....7.2) This characterisation of bisimilarity is used in an associated article [9] to obtain a structural characterisation of classes of polynomial coalgebras de nable by sets of rigid observable formulas. The result is an analogue for polynomial coalgebras of Birkho s famous characterisation in [2] of equational classes of abstract algebras as being those closed under direct products, homomorphic images and subalgebras. The coalgebraic analogue requires the development of a new notion of observational ultrapower of coalgebras. Yet another approach [7] involves a Stone space type of ....

Garret Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

....state dependent. Such Boolean combinations are constrained to be either true or false at each state of a coalgebra. They play a role for polynomial coalgebras analogous to that played by ordinary equations in the theory of abstract algebras. In the latter theory there is a famous result of Birkho [3] characterizing the equationally de nable classes of algebras as being those that are closed under direct products, subalgebras, and homomorphic images. In [6, 4] an analogous structural characterization was obtained for classes of polynomial coalgebras de nable by observable formulas. This ....

Garret Birkho, On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society 31 (1935), 433-454.

....see Chang and Keisler 13 [13] or Hodges [40] The material presented below originated in the context of universal algebra, in which signatures are purely functional, i.e. contain no relation symbols, but can be generalized to allow relations. 2.4. 1 Varieties The following theorem of Birkho [11] predates by about twenty years work in model theory on preservation theorems. Below is actually a generalization allowing relation symbols in the signature (see Hodges [40] Corollary 9.2.8) Theorem 2.4.1. Let be a signature and K a class of structures. Then the following are equivalent. ....

G. Birkho. On the structure of abstract algebras. Proc. Camb. Phil. Soc., 31:433-454, 1935.

....constructors, and to equational term rewriting. They noted that the key property for the technique to succeed was inductive completion, i.e. that the inductive theory of constructors coincides with the equational theory. This was referred to in [KM87] as a Birkoff like theorem, referring to [Bir35] where Birkoff showed the completeness of the rules of inference of equality. Jouannaud and Kounalis proposed to show that this condition could be satisfied with their two key notions. The key theorem of their work was this: If R 0 is a set of rules, and l r is a rule s.t. l is inductively ....

G. Birkoff. On the structure of abstract algebras. In Proceedings of the Cambridge Philosophical Society, volume 29, pages 433--454, 1935.

....and motivate further study of the metatheory of these formulas. In particular we consider the extent to which their role is analogous to the role played by equations in the general theory of universal algebras. Now a cornerstone of classical equational logic is the variety theorem of Birkho [3], stating that a class of algebras is the class of all models of some set of equations i it is closed under homomorphic images, subalgebras, and direct products. This paper aims to prove an analogous result for polynomial coalgebras, giving a structural characterisation of classes of coalgebras ....

Garret Birkho. On the Structure of Abstract Algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

....this area (see section 3 of this paper) To be sure, there are many models of different natures known for Int. A semantics for Int is adequate if Int is (sound and) complete with respect to this semantics. A number of adequate semantics for intuitionistic logic have been found: algebraic (Birkhof, [11]) topological (McKinsey Tarski, 48] Kripke semantics ( 41] and some others. Algebraic models for Int are given by pseudo boolean algebras, which generalizes the boolean algebra semantics of classical logic. Topological semantics for Int is similar to set theoretical semantics for classical ....

G. Birkhof, "On the structure of abstract algebras", Proceedings of the Cambridge Philosophical Society, v.31, pp.433-454, 1935

....of a covariety which is not closed under bisimulations. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1. Introduction One of the earliest theorems in universal algebra is Garrett Birkho s Variety Theorem [Bir35]. It states that a class V of algebras is closed under homomorphic images, subalgebras and products just in case V is the collection of all algebras satisfying some set of equations. The classical de nition of algebras for a signature generalizes to the category theoretic notion of algebras ....

G. Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433 - 454, 1935.

....logic. More technically, they give rise to a fibration with certain structure. The aim of this paper is characterise this structure. The original motivation for the investigations in this paper comes from Birkho# s famous results about definability and deducibility for universal algebras [Bir35]. There has been a considerable amount of work over the last few years aimed at dualizing these results to the setting of co algebras, including [Rut00] GS01] Gum01] Kur01] Kur00] KR02] AH00] Hug01a] Hug02] Ros00] AP01] Successful dualization often requires a reformulation at ....

....etc. are essentially logical theorems. That is, the variety theorem states an equivalence between models of an appropriate logical theory and closure conditions on collections of objects, while the completeness theorem states the analogous result for the logic at hand (equational, in the case of [Bir35]) Our aim is to reinterpret these results in a fibred setting, where the predicates in a fibration give the logic and the types provide the semantics for the language. We present an interpretation of the basic notions of Birkho# s variety theorem in a fibred setting and give a preliminary result ....

G. Birkho#. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433--454, 1935.

....logic. More technically, they give rise to a bration with certain structure. The aim of this paper is characterise this structure. The original motivation for the investigations in this paper comes from Birkho s famous results about de nability and deducibility for universal algebras [Bir35]. There has been a considerable amount of work over the last few years aimed at dualizing these results to the setting of co algebras, including [Rut00] GS01] Gum01] Kur01] Kur00] KR02] AH00] Hug01] Hug02] Ro s00] AP01] Successful dualization often requires a reformulation at ....

....etc. are essentially logical theorems. That is, the variety theorem states an equivalence between models of an appropriate logical theory and closure conditions on collections of objects, while the completeness theorem states the analogous result for the logic at hand (equational, in the case of [Bir35]) Our aim is to reinterpret these results in a bred setting, where the predicates in a bration give the logic and the types provide the semantics for the language. We present an interpretation of the basic notions of Birkho s variety theorem in a bred setting and give a preliminary result ....

G. Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

.... [Rei95, Jac96] Since coalgebras are dual to algebras (in the sense of category theory) this has provided impetus for the study of aspects of coalgebraic theory that correspond to parts of the general theory of algebras [Rut95, Rut00] A notable case is Birkho s celebrated variety theorem [Bir35], stating that a class of algebras is de nable by equations i it is closed under homomorphic images, subalgebras, and direct products. These three constructions dualise to subcoalgebras, images of coalgebraic morphisms and disjoint unions, respectively, and hence a class of coalgebras closed ....

Garret Birkho. On the Structure of Abstract Algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

....rewrite relation E T ( X) T ( X) is de ned as C[s ] E C[t ] for every context C, substitution and equation s t 2 E. E is the symmetric closure of E . The convertibility relation E is the re exive transitive closure of E . 2. 4 Birkho We show Birkho s theorem [3] expressing that semantic and syntactic consequence coincide. To that end we rst construct term models out of term algebras via the intermediate notion of congruence model. De nition 12. A congruence model of an equational speci cation h ; X;Ei consists of a algebra A and a relation on the ....

....and the analysis required and performed is useful. It would be an interesting exercise to give a similar analysis of higher order equational logics [1] comprising higher order rewriting [8] and their termination models [7] Another direction would be to generalize the other well known result of [3]: the characterisation of equational varieties as varieties which are closed under building subalgebras, homomorphic images, and direct products. Acknowledgments Thanks are due to ETL, Tsukuba and the Institute of Information Sciences and Electronics, of Tsububa University for travel support, and ....

G. Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31(4):433-454, 1935.

....# E # T (#, X) T (#, X) is defined as C[s # ] # E C[t # ] for every context C, substitution # and equation s # t # E. # E is the symmetric closure of # E . The convertibility relation # # E is the reflexive transitive closure of # E . 2. 4 Birkho# We show Birkho# s theorem [3] expressing that semantic and syntactic consequence coincide. To that end we first construct term models out of term algebras via the intermediate notion of congruence model. Definition 12. A congruence model of an equational specification ##, X,E# consists of a # algebra A and a relation ....

....and the analysis required and performed is useful. It would be an interesting exercise to give a similar analysis of higher order equational logics [1] comprising higher order rewriting [8] and their termination models [7] Another direction would be to generalize the other well known result of [3]: the characterisation of equational varieties as varieties which are closed under building subalgebras, homomorphic images, and direct products. Acknowledgments Thanks are due to ETL, Tsukuba and the Institute of Information Sciences and Electronics, of Tsububa University for travel support, and ....

G. Birkho#. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31(4):433--454, 1935.

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G. Birkho. On the structure of abstract algebras. In Proceedings of The Cambridge Philosophical Society, volume 31, pages 433-454, 1935.

No context found.

G. Birkho#. On the structure of abstract algebras. In Proceedings of The Cambridge Philosophical Society, volume 31, pages 433--454, 1935.

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G. Birkho. On the structure of abstract algebras. In Proc. Cambridge Philos. Soc., volume 31, pages 417-429, 1935.

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G. Birkho#, On the structure of abstract algebras, Proc. Cambr. Philos. Soc. 31 (1935), 433--454.

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G. Birkho#, On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society, vol. 31 (1935), pp. 433--454.

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G. Birkho#. On the structure of abstract algebras. Proc. Camb. Philos. Soc., 31:433-- 454, 1935.

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Garret Birkho. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31:433-454, 1935.

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G. Birkho#, On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 31 (1935), 433--454.

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G. Birkho#, On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 31 (1935) 433--454.

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G. Birkho: On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society, 31: 433-454, 1935.

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