Amalgamation and interpolation in ordered algebras

The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension property, and the flat amalgamation property and the deductive interpolation property, as well as various dependencies between these properties. These relationships are then exploited in the second part of the paper in order to provide new proofs of amalgamation and deductive interpolation for the varieties of lattice-ordered abelian groups and MV-algebras, and to determine important subvarieties of residuated lattices where these properties hold or fail. In particular, a full description is given of all subvarieties of commutative GMV-algebras possessing the amalgamation property.

Admissibility in Finitely Generated Quasivarieties

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in Q is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

Admissibility via natural dualities

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.

Free abdominal fluid without obvious solid organ injury upon CT imaging: an actual problem or simply over-diagnosing?

Whereas a non-operative approach for hemodynamically stable patients with free intraabdominal fluid in the presence of solid organ injury is generally accepted, the presence of free fluid in the abdomen without evidence of solid organ injury not only presents a challenge for the treating emergency physician but also for the surgeon in charge. Despite recent advances in imaging modalities, with multi-detector computed tomography (CT) (with or without contrast agent) usually the imaging method of choice, diagnosis and interpretation of the results remains difficult. While some studies conclude that CT is highly accurate and relatively specific at diagnosing mesenteric and hollow viscus injury, others studies deem CT to be unreliable. These differences may in part be due to the experience and the interpretation of the radiologist and/or the treating physician or surgeon.A search of the literature has made it apparent that there is no straightforward answer to the question what to do with patients with free intraabdominal fluid on CT scanning but without signs of solid organ injury. In hemodynamically unstable patients, free intraabdominal fluid in the absence of solid organ injury usually mandates immediate surgical intervention. For patients with blunt abdominal trauma and more than just a trace of free intraabdominal fluid or for patients with signs of peritonitis, the threshold for a surgical exploration - preferably by a laparoscopic approach - should be low. Based on the available information, we aim to provide the reader with an overview of the current literature with specific emphasis on diagnostic and therapeutic approaches to this problem and suggest a possible algorithm, which might help with the adequate treatment of such patients.