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15
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
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Cited by 404 (43 self)
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In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 269 (36 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
On free conformal and vertex algebras
 J. Algebra
, 1999
"... Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduc ..."
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Cited by 35 (8 self)
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Vertex algebras and conformal algebras have recently attracted a lot of attention due to their connections with physics and Moonshine representations of the Monster. See, for example, [6], [10], [15], [17], [19]. In this paper we describe bases of free conformal and free vertex algebras (as introduced in [6], see also [20]). All linear spaces are over a field k of characteristic 0. Throughout this paper Z+ will stand for the set of nonnegative integers. In §1 and §2 we give a review of conformal and vertex algebra theory. All statements is these sections are either in [9], [15], [16], [17], [18], [20] or easily follow from results therein. In §3 we investigate free conformal and vertex algebras. 1. Conformal algebras 1.1. Definition of conformal algebras. We first recall some basic definitions and constructions, see [16], [17], [18], [20]. The main object of investigation is defined as follows: Definition 1.1. A Conformal algebra is a linear space C endowed with a linear operator D: C → C and a sequence of bilinear products ○n: C ⊗ C → C, n ∈ Z+, such that for any a, b ∈ C one has (i) (locality) There is a nonnegative integer N = N(a, b) such that a ○n b = 0 for any n � N; (ii) D(a ○n b) = (Da) ○n b + a ○n (Db); (iii) (Da) ○n b = −na n−1 b. 1.2. Spaces of power series. Now let us discuss the main motivation for the Definition 1.1. We closely follow [14] and [18]. 1.2.1. Circle products. Let A be an algebra. Consider the space of power series A[[z, z −1]]. We will write series a ∈ A[[z, z −1]] in the form a(z) = ∑ a(n)z −n−1, a(n) ∈ A. n∈Z On A[[z, z−1]] there is an infinite sequence of bilinear products ○n, n ∈ Z+, given by n a ○n b (z) = Resw a(w)b(z)(z − w) ). (1.1) Explicitly, for a pair of series a(z) = ∑ n∈Z a(n)z−n−1 and b(z) = ∑ n∈Z b(n)z−n−1 we have −m−1 a ○n b (z) = a ○n b (m)z, where
A Calculus of Transition Systems (towards Universal Coalgebra)
 In Alban Ponse, Maarten de Rijke, and Yde Venema, editors, Modal Logic and Process Algebra, CSLI Lecture Notes No
, 1995
"... By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). ..."
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By representing transition systems as coalgebras, the three main ingredients of their theory: coalgebra, homomorphism, and bisimulation, can be seen to be in a precise correspondence to the basic notions of universal algebra: \Sigmaalgebra, homomorphism, and substitutive relation (or congruence). In this paper, some standard results from universal algebra (such as the three isomorphism theorems and facts on the lattices of subalgebras and congruences) are reformulated (using the afore mentioned correspondence) and proved for transition systems. AMS Subject Classification (1991): 68Q10, 68Q55 CR Subject Classification (1991): D.3.1, F.1.2, F.3.2 Keywords & Phrases: Transition system, bisimulation, universal coalgebra, universal algebra, congruence, homomorphism. Note: This paper will appear in `Modal Logic and Process Algebra', edited by Ponse, De Rijke and Venema [PRV95]. 2 Table of Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...
Combinatorics of free vertex algebras
 J. Algebra
"... This paper illustrates the combinatorial approach to vertex algebra — study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds [2] was the first to note that free vertex algebras do not exist in general. The ..."
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This paper illustrates the combinatorial approach to vertex algebra — study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds [2] was the first to note that free vertex algebras do not exist in general. The
FREE ADEQUATE SEMIGROUPS
, 902
"... Abstract. We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup ar ..."
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Abstract. We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial “folding ” operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are Jtrivial and never finitely generated as semigroups, and that those which are finitely generated as (2, 1,1)algebras have decidable word problem. 1.
Secondorder algebraic theories
, 2010
"... Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: ..."
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Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet:
Free left and right adequate semigroups
"... Abstract. Recent research of the author has given an explicit geometric description of free (twosided) adequate semigroups and monoids, as sets of labelled directed trees under a natural combinatorial multiplication. In this paper we show that there are natural embeddings of each free right adequat ..."
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Abstract. Recent research of the author has given an explicit geometric description of free (twosided) adequate semigroups and monoids, as sets of labelled directed trees under a natural combinatorial multiplication. In this paper we show that there are natural embeddings of each free right adequate and free left adequate semigroup or monoid into the corresponding free adequate semigroup or monoid. The corresponding classes of trees are easily described and the resulting geometric representation of free left adequate and free right adequate semigroups is even easier to understand than that in the twosided case. We use it to establish some basic structural properties of free left and right adequate semigroups and monoids. 1.
A Note on a Parameterized Version of the WellFounded Induction Principle
, 1995
"... The wellknown and powerful proof principle by wellfounded induction says that for verifying 8x : P (x) for some property P it suffices to show 8x : [[8y ! x : P (y)] =) P (x)] , provided ! is a wellfounded partial ordering on the domain of interest. Here we investigate a more general formulation ..."
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The wellknown and powerful proof principle by wellfounded induction says that for verifying 8x : P (x) for some property P it suffices to show 8x : [[8y ! x : P (y)] =) P (x)] , provided ! is a wellfounded partial ordering on the domain of interest. Here we investigate a more general formulation of this proof principle which allows for a kind of parameterized partial orderings ! x which naturally arises in some cases. More precisely, we develop conditions under which the parameterized proof principle 8x : [[8y ! x x : P (y)] =) P (x)] is sound in the sense that 8x : [[8y ! x x : P (y)] =) P (x)] =) 8x : P (x) holds, and give counterexamples demonstrating that these conditions are indeed essential. 1 Introduction and Motivation In proofs by wellfounded induction (cf. e.g. [Coh65], [Fef77], [MW93]) one usually tries to verify 8x : P (x) (1) by showing 8x : [ [8 y ! x : P (y) ] =) P (x) ] (2) where ! is a fixed wellfounded partial ordering on the domain of interest. In fact, ! n...