Intuitionistic linear temporal logic (iLTL) has been studied extensively, especially in the last decade. It enjoys natural semantics over intuitionistic Kripke frames equipped with an order-preserving function representing the temporal dynamics, known as 'expanding models'. This leads to a logic that is known to be decidable but whose axiomatisation has long remained open. We propose an extension of iLTL with the co-implication connective of Hilbert–Brouwer logic and call it 'bi-intuitionistic linear temporal logic' (biLTL). We establish that this extension is still decidable for the class of expanding models. We moreover give a sound and complete Hilbert-style calculus for it, the first for any logic extending iLTL. As a corollary, the topological semantics for intuitionistic propositional logic cannot be extended to a topological semantics for Hilbert-Brouwer logic, which thus establishes co-implication as a distinctive feature of the Kripke semantics for bi-intuitionistic logic.

We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.

We investigate intuitionistic logics extended both with the co-implication connective of Hilbert-Brouwer logic and with diamond and box modalities. We use a Kripke semantics based on frames with two 'forth' confluence conditions on the modal relation with respect to the intuitionistic relation. We give sound and strongly complete axiomatisations for entailment on this class of frames, and give similar axiomatisations for the subclasses of frames satisfying any combination of reflexivity, transitivity, and seriality. We then prove that all of these logics are decidable, by proving that they have the finite frame property.

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjoint-union partial algebra of sets but prove that no finite axiomatisation exists. We do the same for other signatures including one or both of disjoint union and subset complement, another partial binary operation we define. Domain-disjoint union is a partial binary operation on partial functions, returning the union if the arguments have disjoint domains and undefined otherwise. For each signature including one or both of domain-disjoint union and subset complement and optionally including composition, we consider the class of partial algebras isomorphic to a collection of partial functions closed under the operations. Again the classes prove to be axiomatisable, but not finitely axiomatisable, in first-order logic. We define the notion of pairwise combinability. For each of the previously considered signatures, we examine the class isomorphic to a partial algebra of sets/partial functions under an isomorphism mapping arbitrary suprema of pairwise combinable sets to the corresponding disjoint unions. We prove that for each case the class is not closed under elementary equivalence. However, when intersection is added to any of the signatures considered, the isomorphism class of the partial algebras of sets is finitely axiomatisable and in each case we give such an axiomatisation.

For representation by partial functions in the signature with intersection, composition and anti-domain, we show that a representation is meet complete if and only if it is join complete. We show that a representation is complete if and only if it is atomic, but that not all atomic representable algebras are completely representable. We show that the class of completely representable algebras is not axiomatizable by any existential-universal-existential first-order theory. By giving an explicit representation, we show that the completely representable algebras form a basic elementary class, axiomatizable by a universal-existential-universal sentence.