Bounded homomorphisms and finitely generated fiber products of lattices
Abstract
We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g:A→D, h:B→D, where A, B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C={(a,b)∈A×B | g(a)=h(b)} is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.
Citation
DeMeo , W , Mayr , P & Ruskuc , N 2020 , ' Bounded homomorphisms and finitely generated fiber products of lattices ' , International Journal of Algebra and Computation , vol. 30 , no. 04 , pp. 693-710 . https://doi.org/10.1142/S0218196720500174
Publication
International Journal of Algebra and Computation
Status
Peer reviewed
ISSN
0218-1967Type
Journal article
Rights
Copyright © 2020 World Scientific Publishing Company. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1142/S0218196720500174
Description
Funding: The first and second authors were supported by the National Science Foundation under Grant No. DMS 1500254.Collections
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