Proof theory for lattice-ordered groups
نویسنده
چکیده
Proof theory can provide useful tools for tackling problems in algebra. In particular, Gentzen systems admitting cut-elimination have been used to establish decidability, complexity, amalgamation, admissibility, and generation results for varieties of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing some family resemblance to groups, such as lattice-ordered groups, MV-algebras, BL-algebras, and cancellative residuated lattices, the proof-theoretic approach has met so far only with limited success. The main aim of this talk will be to introduce proof-theoretic methods for the class of lattice-ordered groups and to explain how these methods can be used to obtain new syntactic proofs of two core theorems: namely, Holland’s result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.
منابع مشابه
A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups
In 1967 Gurevich [3] published a proof that the class of divisible Axchimedean lat t ice-ordered abelian groups such that the lattice of carriers is an atomic Boolean algebra has a hereditarily undecidable first-order theory. (He essentially showed the reduct of this class to lattices has a hereditarily undecidable first-order theory: on p. 49 of his paper change z ~ u + v to z ~ u v v in the d...
متن کاملFinitely generated lattice-ordered groups with soluble word problem
William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be `-embedded in an `-simple lattice-or...
متن کاملConvex $L$-lattice subgroups in $L$-ordered groups
In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...
متن کاملFinitely Presented Abelian Lattice-Ordered Groups
We give necessary and sufficient conditions for the first-order theory of a finitely presented abelian lattice-ordered group to be decidable. We also show that if the number of generators is at most 3, then elementary equivalence implies isomorphism. We deduce from our methods that the theory of the free MV -algebra on at least 2 generators is undecidable.
متن کاملVarieties of Unital `-Groups and ΨMV-Algebras
(Day 1): References (for general background) →Varieties of lattice-ordered groups, N. R. Reilly, in Lattice-Ordered Groups, Advances and Techniques, A. M. W. Glass and W. C. Holland (eds.), Kluwer Academic Publishers, 1989. Lattice-Ordered Groups, an Introduction, M. Anderson and T. Feil, D. Reidel Pub. Co., 1988. Theory of Lattice-Ordered Groups, M. Darnel, Marcel Dekker, 1995. Partially Order...
متن کامل