We axiomatize permutation equivalence in term rewriting systems and Klop’s orthogonal Combinatory Reduction Systems (Klop 1980). The axioms for the former are provided by the general approach proposed by Meseguer (Meseguer 1992). The latter need extra axioms modelling the interplay between reductions and the operation of substitution.

As a consequence of this work, the definition of permutation equivalence is rid of residual calculi, which are heavy in general.

A polynomial-time algorithm is presented for deciding bisimulation equivalence of so-called Basic Parallel Processes: multisets of elementary processes combined by a commutative parallel-composition operator.

An analysis of relationships between Craig-style interpolation, compactness, and other related model-theoretic properties is carried out in the softer framework of categories of pre-institutions. While the equivalence between sentence interpolation and the Robinson property under compactness and Boolean closure is well known, a similar result under different assumptions (not involving compactness) is newly established for presentation interpolation. The standard concept of naturality of model transformation is enriched by a new property, termed restriction adequacy, which proves useful for the reduction of interpolation along pre-institution transformations. A distinct reduction theorem for the Robinson property is presented as well. A variant of the ultraproduct concept is further introduced, and the related closure property for pre-institutions is shown to be equivalent to compactness

Bilattices were introduced and applied by Ginsberg and Fitting for a diversity of applications, such as truth maintenance systems, default inferences and logic programming. In this paper we investigate the structure and properties of a particularly important class of bilattices called interlaced bilattices, which were introduced by Fitting. The main results are that every interlaced bilattice is isomorphic to the Ginsberg-Fitting product of two bounded lattices and that the variety of interlaced bilattices is equivalent to the variety of bounded lattices with two distinguishable distributive elements, which are complements of each other. This implies that interlaced bilattices can be characterized using a finite set of equations. Our results generalize to interlaced bilattices some results of Ginsberg, Fitting and Jónsson for distributive bilattices.

It is shown that the category of fuzzy relations with values in a ∗-autonomous poset gives a ∗-autonomous category. It is also shown that this construction can be generalized by replacing the ∗-autonomous poset by a ∗-autonomous category. In both cases, cofree coalgebras exist so that there is a natural! operation.