In this paper we define the reticulation of a residuated lattice, prove that it has “good properties“, present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.

A number of Arabidopsis (Arabidopsis thaliana) mutants exhibit leaf reticulation, having green veins that stand out against paler interveinal tissues, fewer cells in the interveinal mesophyll, and normal perivascular bundle sheath cells. Here, to examine the basis of leaf reticulation, we analyzed the Arabidopsis RETICULATA-RELATED (RER) gene family, several members of which cause leaf reticula...

In this article we prove a set of preservation properties of the reticulation functor for residuated lattices (for instance preservation of subalgebras, finite direct products, inductive limits, Boolean powers) and we transfer certain properties between bounded distributive lattices and residuated lattices through the reticulation, focusing on Stone, strongly Stone and m-Stone algebras. 2000 Ma...

We describe some new and recent results that allow for the analysis and representation of reticulate evolution by non-tree networks. In particular, we (1) present a simple result to show that, despite the presence of reticulation, there is always a well-defined underlying tree that corresponds to those parts of life that do not have a history of reticulation; (2) describe and apply new theory f...

In phylogenetics, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted arbitrary acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree is contained in a phylogenetic network on the same taxa. ...

In studies of molecular evolution, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree (resp. cluster) is contained in a phylogenetic net...

Processes such as hybridization, horizontal gene transfer, and recombination result in reticulation which can be modeled by phylogenetic networks. Earlier likelihood-based methods for inferring phylogenetic networks from sequence data have been encumbered by the computational challenges related to likelihood evaluations. Consequently, they have required that the possible network hypotheses be g...

Motivation: A major unresolved problem in phylogeny is to reconstruct the evolutionary history of a set of species when it cannot be modeled by a simple tree. When a reticulation event, such as hybrid speciation or horizontal transfer, occurs, separate regions of a species’ genetic data may evolve along different trees. Our approach uses the gene trees for each region to solve the inverse probl...

The DSC method has been employed to monitor the kinetics of reticulation of aqueous polyurethane systems without catalysts, and with the commercial catalyst of zirconium (CATXC-6212) and the highly selective manganese catalyst, the complex Mn(III)diacetylacetonemaleinate (MAM). Among the polyol components, the acrylic emulsions were used for reticulation in this research, and as suitable reticu...