Ing. Matěj Dostál, Ph.D.

It is well known that classical varieties of Sigma-algebras correspond bijectively to finitary monads on Set. We present an analogous result for varieties of ordered Sigma-algebras, that is, categories of algebras presented by inequations between Sigma-terms. We prove that they correspond bijectively to strongly finitary monads on Pos. That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are linings of finitary monads on Set. Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on Set to strongly finitary monads on Pos.

We present a finitary version of Moss' coalgebraic logic for T-coalgebras, where T is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor T-omega(partial derivative) , and the semantics of the modality is given by relation lifting. For the semantics to work, T is required to preserve exact squares. For the finitary setting to work, T-omega(partial derivative) is required to preserve finite intersections. We develop a notion of a base for subobjects of T omega X. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system.

Recent work on compositional distributional models shows that bialgebras over finite dimensional vector spaces can be applied to treat generalised quantifiers for natural language. That technique requires one to construct the vector space over powersets, and therefore is computationally costly. In this paper, we overcome this problem by considering fuzzy versions of quantifiers along the lines of Zadeh, within the category of many valued relations. We show that this category is a concrete instantiation of the compositional distributional model. We show that the semantics obtained in this model is equivalent to the semantics of the fuzzy quantifiers of Zadeh. As a result, we are now able to treat fuzzy quantification without requiring a powerset construction.

Birkhoff's variety theorem from universal algebra characterises equational subcategories of varieties. We give an analogue of Birkhoff's theorem in the setting of enrichment in categories. For a suitable notion of an equational subcategory we characterise these subcategories by their closure properties in the ambient algebraic category.

Morita equivalence detects the situation in which two different theories admit the same class of models for the given theories. We generalise the result of Adamek, Sobral and Sousa concerning Morita equivalence of many-sorted algebraic theories. This generalisation is two-fold. We work in an enriched setting, so the result is parametric in the choice of enrichment. Secondly, the result works for a reasonably general notion of a theory: the class of limits in the theory can be varied. As an example of an application of our result, we show enriched and many-sorted Morita equivalence results, and we recover the known results in the ordinary case.

The notion of relation lifting can be generalised to work with many-valued relations while retaining many vital properties of the “classical” relation lifting. We show that polynomial endofunctors of the category of sets and mappings admit V -relation lifting for relations taking values from a commutative quantale V . Using the technique of functor presentations, we then show that every finitary weak pullback preserving functor admits a V -relation lifting for V being a complete Heyting algebra. As an application of the many-valued lifting we inspect the notion of many-valued bisimulation and we introduce an expressive many-valued variant of Moss’ logic for T-coalgebras, parametric in the functor T.