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14
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 408 (42 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 271 (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 (7 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
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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Cited by 8 (2 self)
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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.
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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Cited by 8 (4 self)
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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
Secondorder algebraic theories
, 2010
"... Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: ..."
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Cited by 6 (5 self)
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Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet:
COHOMOLOGICAL FINITENESS CONDITIONS AND CENTRALISERS IN GENERALISATIONS OF THOMPSON’S Group V
, 2014
"... ..."
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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Cited by 1 (1 self)
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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.
TABLE OF CONTENTS
, 2004
"... This paper examines the level of consistency, content and contextual information of six individual and two collaborative museum databases. A uniform set of elements within several categories was chosen, such as subject crossreferencing and syntax regularity, though some tests were modified to accom ..."
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This paper examines the level of consistency, content and contextual information of six individual and two collaborative museum databases. A uniform set of elements within several categories was chosen, such as subject crossreferencing and syntax regularity, though some tests were modified to accommodate each museum’s differing features. Due to the limited scope of the study and these variations, the conclusions are not intended as definitive judgments. Rather, they provide an outline and evaluation of museum databases and site features and, hopefully, generate ideas for further research.