MINI-COURSE 1: MV-Algebras and l-Groups as Residuated Lattices, directed by
Nick Galatos
Examples of residuated lattices include
lattice-ordered groups, Boolean algebras, Heyting algebras,
MV-algebras and relation algebras. Their common feature is that of a
residuated binary operation on a lattice structure. Residuated
lattices, although encompassing many well known ordered algebras, have
a robust structure theory. At the same time they serve as algebraic
models of many non-classical logics, known as substructural logics,
and are connected even to mathematical linguistics. Substructural
logics include classical, intuitionistic, relevance, linear and fuzzy
logic.
In this mini-course, after presenting a plethora of examples and
discussing their connections (for example, between lattice
ordered-groups and MV-algebras), we will outline the main points of
the general theory of residuated lattices. We will give the logical
systems corresponding to various classes of residuated lattices and
explain the nature of their connection. Special topics will include a
discussion of decidability and of the structure of the subvariety
lattice.
MINI-COURSE 2: Archimedean l-Groups, directed by Warren Wm. McGovern
In this mini-course we will discuss the main tools used in the study of
archimedean l-groups. This will include discussions on (but not limited to) convex
l-subgroups, polar subgroups, prime subgroups, values, the Yosida space of an
element, and the Yosida embedding of an archimedean l-group with weak order unit.
We will also spend some time discussing lattice-ordered groups of continuous functions on
a topological space as well as different types of extensions of lattice-ordered groups.
MINI-COURSE 3: Baker-Beynon duality for lattice-ordered Abelian groups
and MV-algebras, directed by Vincenzo Marra
In this mini-course we will discuss Baker-Beynon duality, that is,
the geometric representation theory of finitely generated projective
Abelian l-groups and finitely presented MV-algebras.
We will develop the necessary tools in piecewise linear topology and
polyhedral geometry as needed. We will further relate Baker-Beynon
duality to the Yosida embedding for Archimedean
l-groups with a weak order unit discussed in Warren McGovern's mini-
course. If time allows, and depending on the participants' interests,
we conclude by discussing applications of
Baker-Beynon duality to current research topics such as measure
theory over unital Abelian l-groups, or decision problems for
finitely presented MV-algebras.
MINI-COURSE 4: Varieties of Pseudo MV-algebras and unital l-groups, directed by
W. Charles Holland
Pseudo MV-algebras are a generalization of MV-algebras which are not necessarily commutative. They are categorically equivalent to not-necessarily commutative unital lattice-ordered groups. That is, they are essentially the same thing, but in completely different languages. This gives new insight into each of these categories.
All of this has considerable overlap with each of the other three mini-courses. In this mini-course we will begin by studying the structure of these algebras, and then we will examine varieties, that is, equationally defined classes of the algebras. There are many relatively recent interesting results, and many open problems that have not yet been carefully looked at.
Workshop on l-groups and MV-algebras
MINI-COURSE 2: Archimedean l-Groups, directed by Warren Wm. McGovern
In this mini-course we will discuss the main tools used in the study of archimedean l-groups. This will include discussions on (but not limited to) convex l-subgroups, polar subgroups, prime subgroups, values, the Yosida space of an element, and the Yosida embedding of an archimedean l-group with weak order unit. We will also spend some time discussing lattice-ordered groups of continuous functions on a topological space as well as different types of extensions of lattice-ordered groups.
MINI-COURSE 3: Baker-Beynon duality for lattice-ordered Abelian groups
and MV-algebras, directed by Vincenzo Marra
In this mini-course we will discuss Baker-Beynon duality, that is, the geometric representation theory of finitely generated projective Abelian l-groups and finitely presented MV-algebras. We will develop the necessary tools in piecewise linear topology and polyhedral geometry as needed. We will further relate Baker-Beynon duality to the Yosida embedding for Archimedean l-groups with a weak order unit discussed in Warren McGovern's mini- course. If time allows, and depending on the participants' interests, we conclude by discussing applications of Baker-Beynon duality to current research topics such as measure theory over unital Abelian l-groups, or decision problems for finitely presented MV-algebras.
MINI-COURSE 4: Varieties of Pseudo MV-algebras and unital l-groups, directed by
W. Charles Holland
Pseudo MV-algebras are a generalization of MV-algebras which are not necessarily commutative. They are categorically equivalent to not-necessarily commutative unital lattice-ordered groups. That is, they are essentially the same thing, but in completely different languages. This gives new insight into each of these categories.
All of this has considerable overlap with each of the other three mini-courses. In this mini-course we will begin by studying the structure of these algebras, and then we will examine varieties, that is, equationally defined classes of the algebras. There are many relatively recent interesting results, and many open problems that have not yet been carefully looked at.
MINI-COURSE 4: Varieties of Pseudo MV-algebras and unital l-groups, directed by
W. Charles Holland
Pseudo MV-algebras are a generalization of MV-algebras which are not necessarily commutative. They are categorically equivalent to not-necessarily commutative unital lattice-ordered groups. That is, they are essentially the same thing, but in completely different languages. This gives new insight into each of these categories.
All of this has considerable overlap with each of the other three mini-courses. In this mini-course we will begin by studying the structure of these algebras, and then we will examine varieties, that is, equationally defined classes of the algebras. There are many relatively recent interesting results, and many open problems that have not yet been carefully looked at.
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