Proof theory for linear lattices
type: Item
instance of: scholarly article
title: Proof theory for linear lattices
zbMATH Open document ID: 0584.06003
DOI: 10.1016/0001-8708(85)90118-5
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: This paper deals with lattices representable as lattices of permuting equivalences (Type I according to Jónsson). Via Mal'cev correspondences it is easily seen that these form a recursively axiomatizable quasivariety, an explicit list of axioms has been given by \textit{B. Jónsson} [Trans. Am. Math. Soc. 92, 449-464 (1959; Zbl 0105.253)]. Gödel's completeness Theorem guarantees the existence of a complete calculus for the universal Horn theory of these lattices, the paper under review shows that the usual method of valuated graphs yields such a calculus. The achievement is to have explicitly formulated this calculus. Thus, proof theoretic results become possible. The author provides a normal form theorem concerning the order of the deductions and a duality theorem for ''planar'' proofs. The reviewer holds that the term ''linear lattices'' should be reserved for lattices of linear subspaces of vector spaces. That these form a class of definitely more special nature is best illustrated by the results of Gel'fand and Ponomarev mentioned in the introduction.
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 03F05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 03G10
zbMATH DE Number: 3935098
zbMATH Keywords: lattices of permuting equivalences
zbMATH Keywords: quasivariety
zbMATH Keywords: axioms
zbMATH Keywords: universal Horn theory
zbMATH Keywords: valuated graphs
zbMATH Keywords: normal form theorem
author: Mark Haiman
reviewed by: Christian Herrmann
MaRDI profile type: Publication
full work available at URL: 0001-8708(85)90118-5
OpenAlex ID: W2003839046
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description: scientific article; zbMATH DE number 3935098