Dragan Mašulović

Towards Weak Bisimulation For Coalgebras

Abstract This paper contains a novel approach to observational equivalence for coalgebras. We describe how to define weak homomorphisms, weak bisimulation, and investigate the connection between them as well as the relation to the known theory of bisimulation for coalgebras. The ultimate result of the paper is the correctness-proof for a weak coinduction proof principle.

ON DUALIZING CLONES AS LAWVERE THEORIES

In this paper we treat clones as Lawvere theories and then dualize them as categories, rather than as single objects of a category of algebras. The approach applies only to some primitive-positive clones, but in return, a structure with somewhat surprising properties is obtained. To illustrate the method, we thoroughly investigate the lattice of clones of operations over a finite boolean algebra.

On finite reflexive homomorphism-homogeneous binary relational systems

A structure is called homogeneous if every isomorphism between finitely induced substructures of the structure extends to an automorphism of the structure. Recently, P.?J.?Cameron and J.?Nesetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely induced substructures of the structure extends to an endomorphism of the structure.In this paper, we consider finite homomorphism-homogeneous relational systems with one reflexive binary relation. We show that for a large part of such relational systems (bidirectionally connected digraphs; a digraph is bidirectionally connected if each of its connected components can be traversed by ? -paths) the problem of deciding whether the system is homomorphism-homogeneous is coNP-complete. Consequently, for this class of relational systems there is no polynomially computable characterization (unless P = N P ). On the other hand, in case of bidirectionally disconnected digraphs we present the full characterization. Our main result states that if a digraph is bidirectionally disconnected, then it is homomorphism-homogeneous if and only if it is either a finite homomorphism-homogeneous quasiorder, or an inflation of a homomorphism-homogeneous digraph with involution (a specific class of digraphs introduced later in the paper), or an inflation of a digraph whose only connected components are C 3 ? and? 1 ? .

Homomorphism-homogeneous monounary algebras

In 2006, P. J. Cameron and J. Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been investigated (e.g. lattices and semilattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasiinjective groups. In this paper we characterize homomorphism-homogenous monounary algebras of arbitrary cardinalities.

Pre-adjunctions and the Ramsey property

Showing that the Ramsey property holds for a class of finite structures can be an extremely challenging task and a slew of sophisticated methods have been proposed in literature. In this paper we propose a new strategy to show that a class of structures has the Ramsey property. The strategy is based on a relatively simple result in category theory and consists of establishing a pre-adjunction between the category of structures which is known to have the Ramsey property, and the category of structures we are interested in. This strategy was implicitly used already in 1981 by H.J. Promel and B. Voigt in their proof of the Ramsey property for the class of finite linearly ordered graphs. We demonstrate the applicability of this strategy by providing short proofs of three important well known results: we show the Ramsey property for the category of all finite linearly ordered posets with embeddings, for the category of finite convexly ordered ultrametric spaces with embeddings, and for the category of all finite linearly ordered metric spaces (rational metric spaces) with embeddings.

Some classes of finite homomorphism-homogeneous point-line geometries

In 2006, P. J. Cameron and J. Nešetřril introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous point-line geometries up to a certain point. We classify all disconnected point-line geometries, and all connected point-line geometries that contain a pair of intersecting proper lines (we say that a line is proper if it contains at least three points). In a way, this is the best one can hope for, since a recent result by Rusinov and Schweitzer implies that there is no polynomially computable characterization of finite connected homomorphism-homogeneous point-line geometries that do not contain a pair of intersecting proper lines (unless P=coNP).

ON THE COMPLEXITY OF DECIDING HOMOMORPHISM-HOMOGENEITY FOR FINITE ALGEBRAS

In 2006, Cameron and Nesetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been described (e.g. monounary algebras and lattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasi-injective groups. In this paper we show that, in general, deciding homomorphism-homogeneity for finite algebras with finitely many fundamental operations and with at least one at least binary fundamental operation is coNP-complete. Therefore, unless P = coNP, there is no feasible characterization of finite homomorphism-homogeneous algebras of this kind.

Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures

For a class of countably infinite ultrahomogeneous structures that generalize edge-colored graphs we provide a probabilistic construction. Also, we give fairly general criteria for the automorphism group of such structures to have the small index property and strong uncountable cofinality, thus generalizing some results of Solecki, Rosendal, and several other authors.

Invariants of monadic coalgebras

Abstract In this paper we consider invariants of computations described by monadic coalgebras, that is, coalgebras for a functor endowed with the structure of a monad. Following the idea of Poschel and Rossiger, we propose another concept of invariants of such coalgebras, namely, the one based on co-relations. We introduce the clone-theoretic apparatus for monadic coalgebras and show that co-relations can be taken for a general representation of their invariants. We then demonstrate that not only subuniverses, but arbitrary λ-simulations can be thought of as invariants of monadic coalgebras, and that the approach to invariants via λ-simulations is inferior in comparison to the one via co-relations. In some cases invariant co-relations uniquely determine the monadic coalgebra. Since the same does not hold in general, to every monadic coalgebra we associate a coalgebra for the same monad which emulates the original one, and has the pleasant property of being uniquely determined by its invariant co-relations.

A universality result for endomorphism monoids of some ultrahomogeneous structures

We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fra¨osse limit) em- beds all countable semigroups. This approach provides us not only with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.