Fall 2023

Lattice-ordered groups are well-studied structures in ordered algebra and they enjoy a Cayley-style representation theorem, which can be used to show that their variety is generated by the order-permutations on the reals and that their equational theory is decidable. Lattice-ordered pregroups are generalizations with two 'inverses' (left and right) and they stem from the study of mathematical linguistics. Distributive lattice-ordered pregroups (DLpG) also enjoy an embedding theorem into functional algebras F(C) on chains C. We improve on this representation by showing that it is enough to consider integral chains: lexicographic products of the integers, and use it to show that the variety of DLpGs is generated by F(Z) and that it has a decidable equational theory. Time permitting, we also discuss the subvarieties of n-periodic  DLpGs, for different naturals n (lattice-ordered groups are exactly the 1-periodic ones) and obtain a generation result for each of them, as well. The decidability of the equational theory is much harder for the periodic case and we obtain it by controlling the size of the associated diagram (a finitary object encapsulating the failure of an equation).

In this talk, I will be presenting results from my Master of Logic thesis “Modal Information Logics”, supervised by Johan van Benthem and Nick Bezhanishvili.

The thesis studies formal properties of a family of so-called modal information logics (MILs)—modal logics first proposed in van Benthem (1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states.

Although MILs have been around for some time, not much is known: van Benthem (2017, 2019) pose two problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability.

The main results of the first part of the talk are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. These results are then build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of Buszkowski (2021) on the Lambek Calculus.

Broadening the study, the basic MIL of suprema on join-semilattices is axiomatized with an infinite scheme. This constitutes the by far most substantive part of the thesis, hence we will only be lending an informal focus to accenting key ideas.

Finally, if time allows, we will comment on the connection between truthmaker semantics and MILs and extend the (compactness and) decidability results in Fine and Jago (2019), chiefly via defining and proving a truthmaker analogue of the FMP.

We build on work by Ploščica (1995) who used digraphs with topology to represent bounded lattices.  The dual digraphs of arbitrary finite lattices were described as TiRS digraphs by Craig, Haviar and Gouveia (2015). Our goal is to characterise the digraphs dual to finite semidistributive and semimodular lattices. We do this by strengthening the defining conditions of TiRS digraphs. Our results lead to a new description of finite convex geometries.

The question of which relation algebras are representable has been studied intensively since the 1940s. More recently, Galatos and Jipsen (2013) described a class of algebras which generalises the class of relation algebras. These so-called quasi relation algebras have a decidable equational theory, but there are no known natural models of binary relations. As a first step in this direction, we will present a definition of representable distributive quasi relation algebras. Much like the case of relation algebras, the definition gives rise to many interesting questions.