14 research outputs found
Order, algebra, and structure: lattice-ordered groups and beyond
This thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics
Proof Theory for Positive Logic with Weak Negation
Proof-theoretic methods are developed for subsystems of Johansson's logic
obtained by extending the positive fragment of intuitionistic logic with weak
negations. These methods are exploited to establish properties of the logical
systems. In particular, cut-free complete sequent calculi are introduced and
used to provide a proof of the fact that the systems satisfy the Craig
interpolation property. Alternative versions of the calculi are later obtained
by means of an appropriate loop-checking history mechanism. Termination of the
new calculi is proved, and used to conclude that the considered logical systems
are PSPACE-complete
Theorems of Alternatives for Substructural Logics
A theorem of alternatives provides a reduction of validity in a substructural
logic to validity in its multiplicative fragment. Notable examples include a
theorem of Arnon Avron that reduces the validity of a disjunction of
multiplicative formulas in the R-mingle logic RM to the validity of a linear
combination of these formulas, and Gordan's theorem for solutions of linear
systems over the real numbers, that yields an analogous reduction for validity
in Abelian logic A. In this paper, general conditions are provided for
axiomatic extensions of involutive uninorm logic without additive constants to
admit a theorem of alternatives. It is also shown that a theorem of
alternatives for a logic can be used to establish (uniform) deductive
interpolation and completeness with respect to a class of dense totally ordered
residuated lattices
Orders on groups, and spectral spaces of lattice-groups
Extending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups
Ordering Free Groups and Validity in Lattice-Ordered Groups
An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (l-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free l-groups and generation of the variety of l-groups by the l-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable l-groups to subsets of relatively free groups that extend to the positive cone of an order
Proof theory for positive logic with weak negation
Proof-theoreticmethods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete