Loading…
From ordered groups to frames
Martínez began in the theory of lattice-ordered groups and found, over a long career at the University of Florida, that many of its questions were really questions about frames. His work on algebraic frames is where the two subjects meet.
The work
Martínez took his doctorate at Tulane University in 1969 under Paul Conrad, one of the founders of the modern theory of lattice-ordered groups. His early papers — on varieties, torsion classes and Archimedean-like classes of ℓ-groups — are in that tradition.
The lattice of convex ℓ-subgroups of an ℓ-group is an algebraic frame: a frame in which every element is a join of compact elements. Martínez saw that this was not a coincidence to be noted and set aside but a structure to be studied. Much of the behaviour of an ℓ-group — its polars, its Archimedean properties, its projectable and Yosida hulls — can be read off from the frame, and the frame-theoretic statement is often cleaner and more general.
Through the 2000s, with Eric Zenk, Warren McGovern, Anthony Hager and others, he developed the theory of algebraic frames in its own right: when such a frame is regular, what dimension means for it, Yosida frames, unit and kernel systems, and a long series on epicompletion in frames with skeletal maps. Frames in which every element is a d-element are now called Martínez frames in his honour.
He was also a builder of community — editor of several volumes from the Ordered Algebraic Structures conferences, and co-editor of the Applied Categorical Structures issue marking Banaschewski's eightieth birthday.
Look at the lattice of convex subgroups long enough and you stop seeing a group. You see a frame.
A life
Born.
Ph.D., Tulane University, supervised by Paul F. Conrad.
Papers on varieties and torsion classes of lattice-ordered groups, from the University of Florida, Gainesville.
Edits proceedings of the Ordered Algebraic Structures conferences in Curaçao and Gainesville.
The algebraic-frames programme: regularity, dimension, Yosida frames, epicompletion.
Co-edits the Applied Categorical Structures issue honouring Banaschewski's 80th birthday.
Dies.
Selected works
Over a hundred papers, roughly split between ℓ-groups and frames. These show the bridge being built.
With E. R. Zenk. Algebra Universalis 50. The basic regularity theory for frames generated by their compact elements.
With E. R. Zenk. Journal of Pure and Applied Algebra 204. Algebraic frames modelled on the Yosida representation of ℓ-groups.
Czechoslovak Mathematical Journal 56. What Krull-style dimension means for an algebraic frame.
Trends in Mathematics. His own history of how frame theory entered the study of ℓ-groups — the best short introduction to why the two subjects belong together.
Algebraic frames sit exactly between the two subjects: frames, but ones whose structure is controlled by algebra. Martínez spent two decades showing how much traffic that bridge can carry.